Theory AOT_PLM

1(*<*)
2theory AOT_PLM
3  imports AOT_Axioms
4begin
5(*>*)
6
7section‹The Deductive System PLM›
8
9(* constrain sledgehammer to the abstraction layer *)
10unbundle AOT_no_atp
11
12AOT_theorem "modus-ponens": assumes φ and φ  ψ shows ψ
13  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
14lemmas MP = "modus-ponens"
15
16AOT_theorem "non-con-thm-thm": assumes  φ shows  φ
17  using assms by simp
18
19AOT_theorem "vdash-properties:1[1]": assumes φ  Λ shows  φ
20  using assms unfolding AOT_model_act_axiom_def by blast (* NOTE: semantics needed *)
21
22text‹Convenience attribute for instantiating modally-fragile axioms.›
23attribute_setup act_axiom_inst =
24  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))
25  "Instantiate modally fragile axiom as modally fragile theorem."
26
27AOT_theorem "vdash-properties:1[2]": assumes φ  Λ shows  φ
28  using assms unfolding AOT_model_axiom_def by blast (* NOTE: semantics needed *)
29
30text‹Convenience attribute for instantiating modally-strict axioms.›
31attribute_setup axiom_inst =
32  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[2]"})))
33  "Instantiate axiom as theorem."
34
35text‹Convenience methods and theorem sets for applying "cqt:2".›
36method cqt_2_lambda_inst_prover = (fast intro: AOT_instance_of_cqt_2_intro)
37method "cqt:2[lambda]" = (rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
38lemmas "cqt:2" = "cqt:2[const_var]"[axiom_inst] "cqt:2[lambda]"[axiom_inst] AOT_instance_of_cqt_2_intro
39method "cqt:2" = (safe intro!: "cqt:2")
40
41AOT_theorem "vdash-properties:3": assumes  φ shows Γ  φ
42  using assms by blast
43
44AOT_theorem "vdash-properties:5": assumes Γ1  φ and Γ2  φ  ψ shows Γ1, Γ2  ψ
45  using MP assms by blast
46
47AOT_theorem "vdash-properties:6": assumes φ and φ  ψ shows ψ
48  using MP assms by blast
49
50AOT_theorem "vdash-properties:8": assumes Γ  φ and φ  ψ shows Γ  ψ
51  using assms by argo
52
53AOT_theorem "vdash-properties:9": assumes φ shows ψ  φ
54  using MP "pl:1"[axiom_inst] assms by blast
55
56AOT_theorem "vdash-properties:10": assumes φ  ψ and φ shows ψ
57  using MP assms by blast
58lemmas "→E" = "vdash-properties:10"
59
60AOT_theorem "rule-gen": assumes for arbitrary α: φ{α} shows α φ{α}
61  using assms by (metis AOT_var_of_term_inverse AOT_sem_denotes AOT_sem_forall) (* NOTE: semantics needed *)
62lemmas GEN = "rule-gen"
63
64AOT_theorem "RN[prem]": assumes Γ  φ shows Γ  φ
65  by (meson AOT_sem_box assms image_iff) (* NOTE: semantics needed *)
66AOT_theorem RN: assumes  φ shows φ
67  using "RN[prem]" assms by blast
68
69AOT_axiom "df-rules-formulas[1]": assumes φ df ψ shows φ  ψ
70  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
71AOT_axiom "df-rules-formulas[2]": assumes φ df ψ shows ψ  φ
72  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
73(* NOTE: for convenience also state the above as regular theorems *)
74AOT_theorem "df-rules-formulas[3]": assumes φ df ψ shows φ  ψ
75  using "df-rules-formulas[1]"[axiom_inst, OF assms].
76AOT_theorem "df-rules-formulas[4]": assumes φ df ψ shows ψ  φ
77  using "df-rules-formulas[2]"[axiom_inst, OF assms].
78
79
80AOT_axiom "df-rules-terms[1]":
81  assumes τ{α1...αn} =df σ{α1...αn}
82  shows (σ{τ1...τn}  τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}  ¬τ{τ1...τn})
83  using assms by (simp add: AOT_model_axiomI AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
84AOT_axiom "df-rules-terms[2]":
85  assumes τ =df σ
86  shows (σ  τ = σ) & (¬σ  ¬τ)
87  by (metis "df-rules-terms[1]" case_unit_Unity assms)
88(* NOTE: for convenience also state the above as regular theorems *)
89AOT_theorem "df-rules-terms[3]":
90  assumes τ{α1...αn} =df σ{α1...αn}
91  shows (σ{τ1...τn}  τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}  ¬τ{τ1...τn})
92  using "df-rules-terms[1]"[axiom_inst, OF assms].
93AOT_theorem "df-rules-terms[4]":
94  assumes τ =df σ
95  shows (σ  τ = σ) & (¬σ  ¬τ)
96  using "df-rules-terms[2]"[axiom_inst, OF assms].
97
98
99AOT_theorem "if-p-then-p": φ  φ
100  by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)
101
102AOT_theorem "deduction-theorem": assumes φ  ψ shows φ  ψ
103  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
104lemmas CP = "deduction-theorem"
105lemmas "→I" = "deduction-theorem"
106
107AOT_theorem "ded-thm-cor:1": assumes Γ1  φ  ψ and Γ2  ψ  χ shows Γ1, Γ2  φ  χ
108  using "→E" "→I" assms by blast
109AOT_theorem "ded-thm-cor:2": assumes Γ1  φ  (ψ  χ) and Γ2  ψ shows Γ1, Γ2  φ  χ
110  using "→E" "→I" assms by blast
111
112AOT_theorem "ded-thm-cor:3": assumes φ  ψ and ψ  χ shows φ  χ
113  using "→E" "→I" assms by blast
114declare "ded-thm-cor:3"[trans]
115AOT_theorem "ded-thm-cor:4": assumes φ  (ψ  χ) and ψ shows φ  χ
116  using "→E" "→I" assms by blast
117
118lemmas "Hypothetical Syllogism" = "ded-thm-cor:3"
119
120AOT_theorem "useful-tautologies:1": ¬¬φ  φ
121  by (metis "pl:3"[axiom_inst] "→I" "Hypothetical Syllogism")
122AOT_theorem "useful-tautologies:2": φ  ¬¬φ
123  by (metis "pl:3"[axiom_inst] "→I" "ded-thm-cor:4")
124AOT_theorem "useful-tautologies:3": ¬φ  (φ  ψ)
125  by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "→I")
126AOT_theorem "useful-tautologies:4": (¬ψ  ¬φ)  (φ  ψ)
127  by (meson "pl:3"[axiom_inst] "Hypothetical Syllogism" "→I")
128AOT_theorem "useful-tautologies:5": (φ  ψ)  (¬ψ  ¬φ)
129  by (metis "useful-tautologies:4" "Hypothetical Syllogism" "→I")
130
131AOT_theorem "useful-tautologies:6": (φ  ¬ψ)  (ψ  ¬φ)
132  by (metis "→I" MP "useful-tautologies:4")
133
134AOT_theorem "useful-tautologies:7": (¬φ  ψ)  (¬ψ  φ)
135  by (metis "→I" MP "useful-tautologies:3" "useful-tautologies:5")
136
137AOT_theorem "useful-tautologies:8": φ  (¬ψ  ¬(φ  ψ))
138  by (metis "→I" MP "useful-tautologies:5")
139
140AOT_theorem "useful-tautologies:9": (φ  ψ)  ((¬φ  ψ)  ψ)
141  by (metis "→I" MP "useful-tautologies:6")
142
143AOT_theorem "useful-tautologies:10": (φ  ¬ψ)  ((φ  ψ)  ¬φ)
144  by (metis "→I" MP "pl:3"[axiom_inst])
145
146AOT_theorem "dn-i-e:1": assumes φ shows ¬¬φ
147  using MP "useful-tautologies:2" assms by blast
148lemmas "¬¬I" = "dn-i-e:1"
149AOT_theorem "dn-i-e:2": assumes ¬¬φ shows φ
150  using MP "useful-tautologies:1" assms by blast
151lemmas "¬¬E" = "dn-i-e:2"
152
153AOT_theorem "modus-tollens:1": assumes φ  ψ and ¬ψ shows ¬φ
154  using MP "useful-tautologies:5" assms by blast
155AOT_theorem "modus-tollens:2": assumes φ  ¬ψ and ψ shows ¬φ
156  using "¬¬I" "modus-tollens:1" assms by blast
157lemmas MT = "modus-tollens:1" "modus-tollens:2"
158
159AOT_theorem "contraposition:1[1]": assumes φ  ψ shows ¬ψ  ¬φ
160  using "→I" MT(1) assms by blast
161AOT_theorem "contraposition:1[2]": assumes ¬ψ  ¬φ shows φ  ψ
162  using "→I" "¬¬E" MT(2) assms by blast
163
164AOT_theorem "contraposition:2": assumes φ  ¬ψ shows ψ  ¬φ
165  using "→I" MT(2) assms by blast
166
167(* TODO: this is actually a mixture of the two variants given in PLM; adjust. *)
168AOT_theorem "reductio-aa:1":
169  assumes ¬φ  ¬ψ and ¬φ  ψ shows φ
170  using "→I" "¬¬E" MT(2) assms by blast
171AOT_theorem "reductio-aa:2":
172  assumes φ  ¬ψ and φ  ψ shows ¬φ
173  using "reductio-aa:1" assms by blast
174lemmas "RAA" = "reductio-aa:1" "reductio-aa:2"
175
176AOT_theorem "exc-mid": φ  ¬φ
177  using "df-rules-formulas[4]" "if-p-then-p" MP "conventions:2" by blast
178
179AOT_theorem "non-contradiction": ¬(φ & ¬φ)
180  using "df-rules-formulas[3]" MT(2) "useful-tautologies:2" "conventions:1" by blast
181
182AOT_theorem "con-dis-taut:1": (φ & ψ)  φ
183  by (meson "→I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
184AOT_theorem "con-dis-taut:2": (φ & ψ)  ψ
185  by (metis "→I" "df-rules-formulas[3]" MT(2) RAA(2) "¬¬E" "conventions:1")
186lemmas "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"
187
188AOT_theorem "con-dis-taut:3": φ  (φ  ψ)
189  by (meson "contraposition:1[2]" "df-rules-formulas[4]" MP "→I" "conventions:2")
190AOT_theorem "con-dis-taut:4": ψ  (φ  ψ)
191  using "Hypothetical Syllogism" "df-rules-formulas[4]" "pl:1"[axiom_inst] "conventions:2" by blast
192lemmas "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"
193
194AOT_theorem "con-dis-taut:5": φ  (ψ  (φ & ψ))
195  by (metis "contraposition:2" "Hypothetical Syllogism" "→I" "df-rules-formulas[4]" "conventions:1")
196lemmas Adjunction = "con-dis-taut:5"
197
198AOT_theorem "con-dis-taut:6": (φ & φ)  φ
199  by (metis Adjunction "→I" "df-rules-formulas[4]" MP "Conjunction Simplification"(1) "conventions:3")
200lemmas "Idempotence of &" = "con-dis-taut:6"
201
202AOT_theorem "con-dis-taut:7": (φ  φ)  φ
203proof -
204  {
205    AOT_assume φ  φ
206    AOT_hence ¬φ  φ
207      using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
208    AOT_hence φ using "if-p-then-p" RAA(1) MP by blast
209  }
210  moreover {
211    AOT_assume φ
212    AOT_hence φ  φ using "Disjunction Addition"(1) MP by blast
213  }
214  ultimately AOT_show (φ  φ)  φ
215    using "conventions:3"[THEN "df-rules-formulas[4]"] MP
216    by (metis Adjunction "→I")
217qed
218lemmas "Idempotence of ∨" = "con-dis-taut:7"
219
220
221AOT_theorem "con-dis-i-e:1": assumes φ and ψ shows φ & ψ
222  using Adjunction MP assms by blast
223lemmas "&I" = "con-dis-i-e:1"
224
225AOT_theorem "con-dis-i-e:2:a": assumes φ & ψ shows φ
226  using "Conjunction Simplification"(1) MP assms by blast
227AOT_theorem "con-dis-i-e:2:b": assumes φ & ψ shows ψ
228  using "Conjunction Simplification"(2) MP assms by blast
229lemmas "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"
230
231AOT_theorem "con-dis-i-e:3:a": assumes φ shows φ  ψ
232  using "Disjunction Addition"(1) MP assms by blast
233AOT_theorem "con-dis-i-e:3:b": assumes ψ shows φ  ψ
234  using "Disjunction Addition"(2) MP assms by blast
235AOT_theorem "con-dis-i-e:3:c": assumes φ  ψ and φ  χ and ψ  Θ shows χ  Θ
236  by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2) "df-rules-formulas[3]" MT(1) RAA(1) "conventions:2" assms)
237lemmas "∨I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"
238
239AOT_theorem "con-dis-i-e:4:a": assumes φ  ψ and φ  χ and ψ  χ shows χ
240  by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
241AOT_theorem "con-dis-i-e:4:b": assumes φ  ψ and ¬φ shows ψ
242  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
243AOT_theorem "con-dis-i-e:4:c": assumes φ  ψ and ¬ψ shows φ
244  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
245lemmas "∨E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"
246
247AOT_theorem "raa-cor:1": assumes ¬φ  ψ & ¬ψ shows φ
248  using "&E" "∨E"(3) "∨I"(2) RAA(2) assms by blast
249AOT_theorem "raa-cor:2": assumes φ  ψ & ¬ψ shows ¬φ
250  using "raa-cor:1" assms by blast
251AOT_theorem "raa-cor:3": assumes φ and ¬ψ  ¬φ shows ψ
252  using RAA assms by blast
253AOT_theorem "raa-cor:4": assumes ¬φ and ¬ψ  φ shows ψ
254  using RAA assms by blast
255AOT_theorem "raa-cor:5": assumes φ and ψ  ¬φ shows ¬ψ
256  using RAA assms by blast
257AOT_theorem "raa-cor:6": assumes ¬φ and ψ  φ shows ¬ψ
258  using RAA assms by blast
259
260(* TODO: note these need manual introduction rules *)
261AOT_theorem "oth-class-taut:1:a": (φ  ψ)  ¬(φ & ¬ψ)
262  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
263     (metis "&E" "&I" "raa-cor:3" "→I" MP)
264AOT_theorem "oth-class-taut:1:b": ¬(φ  ψ)  (φ & ¬ψ)
265  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
266     (metis "&E" "&I" "raa-cor:3" "→I" MP)
267AOT_theorem "oth-class-taut:1:c": (φ  ψ)  (¬φ  ψ)
268  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
269     (metis "&I" "∨I"(1, 2) "∨E"(3) "→I" MP "raa-cor:1")
270
271AOT_theorem "oth-class-taut:2:a": (φ & ψ)  (ψ & φ)
272  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
273     (meson "&I" "&E" "→I")
274lemmas "Commutativity of &" = "oth-class-taut:2:a"
275AOT_theorem "oth-class-taut:2:b": (φ & (ψ & χ))  ((φ & ψ) & χ)
276  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
277     (metis "&I" "&E" "→I")
278lemmas "Associativity of &" = "oth-class-taut:2:b"
279AOT_theorem "oth-class-taut:2:c": (φ  ψ)  (ψ  φ)
280  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
281     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
282lemmas "Commutativity of ∨" = "oth-class-taut:2:c"
283AOT_theorem "oth-class-taut:2:d": (φ  (ψ  χ))  ((φ  ψ)  χ)
284  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
285     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
286lemmas "Associativity of ∨" = "oth-class-taut:2:d"
287AOT_theorem "oth-class-taut:2:e": (φ  ψ)  (ψ  φ)
288  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I";
289      metis "&I" "df-rules-formulas[4]" "conventions:3" "&E" "Hypothetical Syllogism" "→I" "df-rules-formulas[3]")
290lemmas "Commutativity of ≡" = "oth-class-taut:2:e"
291AOT_theorem "oth-class-taut:2:f": (φ  (ψ  χ))  ((φ  ψ)  χ)
292  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
293        "→I" "→E" "&E" "&I"
294  by metis
295lemmas "Associativity of ≡" = "oth-class-taut:2:f"
296
297AOT_theorem "oth-class-taut:3:a": φ  φ
298  using "&I" "vdash-properties:6" "if-p-then-p" "df-rules-formulas[4]" "conventions:3" by blast
299AOT_theorem "oth-class-taut:3:b": φ  ¬¬φ
300  using "&I" "useful-tautologies:1" "useful-tautologies:2" "vdash-properties:6" "df-rules-formulas[4]" "conventions:3" by blast
301AOT_theorem "oth-class-taut:3:c": ¬(φ  ¬φ)
302  by (metis "&E" "→E" RAA "df-rules-formulas[3]" "conventions:3")
303
304AOT_theorem "oth-class-taut:4:a": (φ  ψ)  ((ψ  χ)  (φ  χ))
305  by (metis "→E" "→I")
306AOT_theorem "oth-class-taut:4:b": (φ  ψ)  (¬φ  ¬ψ)
307  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
308        "→I" "→E" "&E" "&I" RAA by metis
309AOT_theorem "oth-class-taut:4:c": (φ  ψ)  ((φ  χ)  (ψ  χ))
310  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
311        "→I" "→E" "&E" "&I" by metis
312AOT_theorem "oth-class-taut:4:d": (φ  ψ)  ((χ  φ)  (χ  ψ))
313  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
314        "→I" "→E" "&E" "&I" by metis
315AOT_theorem "oth-class-taut:4:e": (φ  ψ)  ((φ & χ)  (ψ & χ))
316  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
317        "→I" "→E" "&E" "&I" by metis
318AOT_theorem "oth-class-taut:4:f": (φ  ψ)  ((χ & φ)  (χ & ψ))
319  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
320        "→I" "→E" "&E" "&I" by metis
321AOT_theorem "oth-class-taut:4:g": (φ  ψ)  ((φ & ψ)  (¬φ & ¬ψ))
322proof(safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"] "&I" "→I"
323           dest!: "conventions:3"[THEN "df-rules-formulas[3]", THEN "→E"])
324  AOT_show φ & ψ  (¬φ & ¬ψ) if (φ  ψ) & (ψ  φ)
325    using "&E" "∨I" "→E" "&I" "raa-cor:1" "→I" "∨E" that by metis
326next
327  AOT_show ψ if φ & ψ  (¬φ & ¬ψ) and φ
328    using that "∨E" "&E" "raa-cor:3" by blast
329next
330  AOT_show φ if φ & ψ  (¬φ & ¬ψ) and ψ
331    using that "∨E" "&E" "raa-cor:3" by blast
332qed
333AOT_theorem "oth-class-taut:4:h": ¬(φ  ψ)  ((φ & ¬ψ)  (¬φ & ψ))
334proof (safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"] "&I" "→I")
335  AOT_show φ & ¬ψ  (¬φ & ψ) if ¬(φ  ψ)
336    by (metis that "&I" "∨I"(1, 2) "→I" MT(1) "df-rules-formulas[4]" "raa-cor:3" "conventions:3")
337next
338  AOT_show ¬(φ  ψ) if φ & ¬ψ  (¬φ & ψ)
339    by (metis that "&E" "∨E"(2) "→E" "df-rules-formulas[3]" "raa-cor:3" "conventions:3")
340qed
341AOT_theorem "oth-class-taut:5:a": (φ & ψ)  ¬(¬φ  ¬ψ)
342  using "conventions:3"[THEN "df-rules-formulas[4]"]
343        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
344AOT_theorem "oth-class-taut:5:b": (φ  ψ)  ¬(¬φ & ¬ψ)
345  using "conventions:3"[THEN "df-rules-formulas[4]"]
346        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
347AOT_theorem "oth-class-taut:5:c": ¬(φ & ψ)  (¬φ  ¬ψ)
348  using "conventions:3"[THEN "df-rules-formulas[4]"]
349        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
350AOT_theorem "oth-class-taut:5:d": ¬(φ  ψ)  (¬φ & ¬ψ)
351  using "conventions:3"[THEN "df-rules-formulas[4]"]
352        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
353
354lemmas DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"
355
356AOT_theorem "oth-class-taut:6:a": (φ & (ψ  χ))  ((φ & ψ)  (φ & χ))
357  using "conventions:3"[THEN "df-rules-formulas[4]"]
358        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
359AOT_theorem "oth-class-taut:6:b": (φ  (ψ & χ))  ((φ  ψ) & (φ  χ))
360  using "conventions:3"[THEN "df-rules-formulas[4]"]
361        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
362
363AOT_theorem "oth-class-taut:7:a": ((φ & ψ)  χ)  (φ  (ψ  χ))
364  by (metis "&I" "→E" "→I")
365lemmas Exportation = "oth-class-taut:7:a"
366AOT_theorem "oth-class-taut:7:b": (φ  (ψ χ))  ((φ & ψ)  χ)
367  by (metis "&E" "→E" "→I")
368lemmas Importation = "oth-class-taut:7:b"
369
370AOT_theorem "oth-class-taut:8:a": (φ  (ψ  χ))  (ψ  (φ  χ))
371  using "conventions:3"[THEN "df-rules-formulas[4]"] "→I" "→E" "&E" "&I" by metis
372lemmas Permutation = "oth-class-taut:8:a"
373AOT_theorem "oth-class-taut:8:b": (φ  ψ)  ((φ  χ)  (φ  (ψ & χ)))
374  by (metis "&I" "→E" "→I")
375lemmas Composition = "oth-class-taut:8:b"
376AOT_theorem "oth-class-taut:8:c": (φ  χ)  ((ψ  χ)  ((φ  ψ)  χ))
377  by (metis "∨E"(2) "→E" "→I" RAA(1))
378AOT_theorem "oth-class-taut:8:d": ((φ  ψ) & (χ  Θ))  ((φ & χ)  (ψ & Θ))
379  by (metis "&E" "&I" "→E" "→I")
380lemmas "Double Composition" = "oth-class-taut:8:d"
381AOT_theorem "oth-class-taut:8:e": ((φ & ψ)  (φ & χ))  (φ  (ψ  χ))
382  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
383            "→I" "→E" "&E" "&I")
384AOT_theorem "oth-class-taut:8:f": ((φ & ψ)  (χ & ψ))  (ψ  (φ  χ))
385  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
386            "→I" "→E" "&E" "&I")
387AOT_theorem "oth-class-taut:8:g": (ψ  χ)  ((φ  ψ)  (φ  χ))
388  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
389            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
390AOT_theorem "oth-class-taut:8:h": (ψ  χ)  ((ψ  φ)  (χ  φ))
391  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
392            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
393AOT_theorem "oth-class-taut:8:i": (φ  (ψ & χ))  (ψ  (φ  χ))
394  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
395            "→I" "→E" "&E" "&I")
396
397AOT_theorem "intro-elim:1": assumes φ  ψ and φ  χ and ψ  Θ shows χ  Θ
398  by (metis assms "∨I"(1, 2) "∨E"(1) "conventions:3"[THEN "df-rules-formulas[3]"] "→I" "→E" "&E"(1))
399
400AOT_theorem "intro-elim:2": assumes φ  ψ and ψ  φ shows φ  ψ
401  by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
402lemmas "≡I" = "intro-elim:2"
403
404AOT_theorem "intro-elim:3:a": assumes φ  ψ and φ shows ψ
405  by (metis "∨I"(1) "→I" "∨E"(1) "intro-elim:1" assms)
406AOT_theorem "intro-elim:3:b": assumes φ  ψ and ψ shows φ
407  using "intro-elim:3:a" "Commutativity of ≡" assms by blast
408AOT_theorem "intro-elim:3:c": assumes φ  ψ and ¬φ shows ¬ψ
409  using "intro-elim:3:b" "raa-cor:3" assms by blast
410AOT_theorem "intro-elim:3:d": assumes φ  ψ and ¬ψ shows ¬φ
411  using "intro-elim:3:a" "raa-cor:3" assms by blast
412AOT_theorem "intro-elim:3:e": assumes φ  ψ and ψ  χ shows φ  χ
413  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
414declare "intro-elim:3:e"[trans]
415AOT_theorem "intro-elim:3:f": assumes φ  ψ and φ  χ shows χ  ψ
416  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
417lemmas "≡E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c" "intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"
418
419declare "Commutativity of ≡"[THEN "≡E"(1), sym]
420
421AOT_theorem "rule-eq-df:1": assumes φ df ψ shows φ  ψ
422  by (simp add: "≡I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
423lemmas "≡Df" = "rule-eq-df:1"
424AOT_theorem "rule-eq-df:2": assumes φ df ψ and φ shows ψ
425  using "≡Df" "≡E"(1) assms by blast
426lemmas "≡dfE" = "rule-eq-df:2"
427AOT_theorem "rule-eq-df:3": assumes φ df ψ and ψ shows φ
428  using "≡Df" "≡E"(2) assms by blast
429lemmas "≡dfI" = "rule-eq-df:3"
430
431AOT_theorem  "df-simplify:1": assumes φ  (ψ & χ) and ψ shows φ  χ
432  by (metis "&E"(2) "&I" "≡E"(1, 2) "≡I" "→I" assms)
433(* TODO: this is a slight variation from PLM *)
434AOT_theorem  "df-simplify:2": assumes φ  (ψ & χ) and χ shows φ  ψ
435  by (metis "&E"(1) "&I" "≡E"(1, 2) "≡I" "→I" assms)
436lemmas "≡S" = "df-simplify:1"  "df-simplify:2"
437
438AOT_theorem "rule-ui:1": assumes α φ{α} and τ shows φ{τ}
439  using "→E" "cqt:1"[axiom_inst] assms by blast
440AOT_theorem "rule-ui:2[const_var]": assumes α φ{α} shows φ{β}
441  by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
442(* TODO: precise proviso in PLM *)
443AOT_theorem "rule-ui:2[lambda]":
444  assumes F φ{F} and INSTANCE_OF_CQT_2(ψ)
445  shows φ{ν1...νn ψ{ν1...νn}]}
446  by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
447AOT_theorem "rule-ui:3": assumes α φ{α} shows φ{α}
448  by (simp add: "rule-ui:2[const_var]" assms)
449lemmas "∀E" = "rule-ui:1" "rule-ui:2[const_var]" "rule-ui:2[lambda]" "rule-ui:3"
450
451AOT_theorem "cqt-orig:1[const_var]": α φ{α}  φ{β} by (simp add: "∀E"(2) "→I")
452AOT_theorem "cqt-orig:1[lambda]":
453  assumes INSTANCE_OF_CQT_2(ψ)
454  shows F φ{F}  φ{ν1...νn ψ{ν1...νn}]}
455  by (simp add: "∀E"(3) "→I" assms)
456AOT_theorem "cqt-orig:2": α (φ  ψ{α})  (φ  α ψ{α})
457  by (metis "→I" GEN "vdash-properties:6" "∀E"(4))
458AOT_theorem "cqt-orig:3": α φ{α}  φ{α} using "cqt-orig:1[const_var]" .
459
460(* TODO: work out difference to GEN *)
461AOT_theorem universal: assumes for arbitrary β: φ{β} shows α φ{α}
462  using GEN assms .
463lemmas "∀I" = universal
464
465(* Generalized mechanism for "∀I" followed by ∀E *)
466ML467fun get_instantiated_allI ctxt varname thm = let
468val trm = Thm.concl_of thm
469val trm = case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
470                      | _ => raise Term.TERM ("Expected simple theorem.", [trm])
471fun extractVars (Const (const_name‹AOT_term_of_var›, _) $ Var v) =
472    (if fst (fst v) = fst varname then [Var v] else []) (* TODO: care about the index? *)
473  | extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
474  | extractVars (Abs (_, _, t)) = extractVars t
475  | extractVars _ = []
476val vars = extractVars trm
477val vars = fold Term.add_vars vars []
478val var = hd vars
479val trmty = case (snd var) of (Type (type_name‹AOT_var›, [t])) => (t)
480              | _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
481val trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
482      Const (const_name‹AOT_term_of_var›, Type ("fun", [snd var, trmty]))
483       $ Var var, trm))
484val trm = Thm.cterm_of (Context.proof_of ctxt) trm
485val ty = hd (Term.add_tvars (Thm.prop_of @{thm "∀I"}) [])
486val typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
487val allthm = Drule.instantiate_normalize ([(ty, typ)],[]) @{thm "∀I"}
488val phi = hd (Term.add_vars (Thm.prop_of allthm) [])
489val allthm = Drule.instantiate_normalize ([],[(phi,trm)]) allthm
490in
491allthm
492end
493
494
495attribute_setup "∀I" =
496  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
497  (fn ctxt => fn thm => fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm))
498  "Quantify over a variable in a theorem using GEN."
499
500attribute_setup "unvarify" =
501  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
502  (fn ctxt => fn thm =>
503    let
504    val thm = fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm
505    val thm = fold (fn _ => fn thm => thm RS @{thm "∀E"(1)}) args thm
506    in
507     thm
508    end))
509  "Generalize a statement about variables to a statement about denoting terms."
510
511(* TODO: rereplace-lem does not apply to the embedding *)
512
513AOT_theorem "cqt-basic:1": αβ φ{α,β}  βα φ{α,β}
514  by (metis "≡I" "∀E"(2) "∀I" "→I")
515
516AOT_theorem "cqt-basic:2": α(φ{α}  ψ{α})  (α(φ{α}  ψ{α}) & α(ψ{α}  φ{α}))
517proof (rule "≡I"; rule "→I")
518  AOT_assume α(φ{α}  ψ{α})
519  AOT_hence φ{α}  ψ{α} for α using "∀E"(2) by blast
520  AOT_hence φ{α}  ψ{α} and ψ{α}  φ{α} for α
521    using "≡E"(1,2) "→I" by blast+
522  AOT_thus α(φ{α}  ψ{α}) & α(ψ{α}  φ{α})
523    by (auto intro: "&I" "∀I")
524next
525  AOT_assume α(φ{α}  ψ{α}) & α(ψ{α}  φ{α})
526  AOT_hence φ{α}  ψ{α} and ψ{α}  φ{α} for α
527    using "∀E"(2) "&E" by blast+
528  AOT_hence φ{α}  ψ{α} for α
529    using "≡I" by blast
530  AOT_thus α(φ{α}  ψ{α}) by (auto intro: "∀I")
531qed
532
533AOT_theorem "cqt-basic:3": α(φ{α}  ψ{α})  (α φ{α}  α ψ{α})
534proof(rule "→I")
535  AOT_assume α(φ{α}  ψ{α})
536  AOT_hence 1: φ{α}  ψ{α} for α using "∀E"(2) by blast
537  {
538    AOT_assume α φ{α}
539    AOT_hence α ψ{α} using 1 "∀I" "∀E"(4) "≡E" by metis
540  }
541  moreover {
542    AOT_assume α ψ{α}
543    AOT_hence α φ{α} using 1 "∀I" "∀E"(4) "≡E" by metis
544  }
545  ultimately AOT_show α φ{α}  α ψ{α}
546    using "≡I" "→I" by auto
547qed
548
549AOT_theorem "cqt-basic:4": α(φ{α} & ψ{α})  (α φ{α} & α ψ{α})
550proof(rule "→I")
551  AOT_assume 0: α(φ{α} & ψ{α})
552  AOT_have φ{α} and ψ{α} for α using "∀E"(2) 0 "&E" by blast+
553  AOT_thus α φ{α} & α ψ{α}
554    by (auto intro: "∀I" "&I")
555qed
556
557AOT_theorem "cqt-basic:5": (α1...∀αn(φ{α1...αn}))  φ{α1...αn}
558  using "cqt-orig:3" by blast
559
560AOT_theorem "cqt-basic:6": αα φ{α}  α φ{α}
561  by (meson "≡I" "→I" GEN "cqt-orig:1[const_var]")
562
563AOT_theorem "cqt-basic:7": (φ  α ψ{α})  α(φ  ψ{α})
564  by (metis "→I" "vdash-properties:6" "rule-ui:3" "≡I" GEN)
565
566AOT_theorem "cqt-basic:8": (α φ{α}  α ψ{α})  α (φ{α}  ψ{α})
567  by (simp add: "∨I"(3) "→I" GEN "cqt-orig:1[const_var]")
568
569AOT_theorem "cqt-basic:9": (α (φ{α}  ψ{α}) & α (ψ{α}  χ{α}))  α(φ{α}  χ{α})
570proof -
571  {
572    AOT_assume α (φ{α}  ψ{α})
573    moreover AOT_assume α (ψ{α}  χ{α})
574    ultimately AOT_have φ{α}  ψ{α} and ψ{α}  χ{α} for α using "∀E" by blast+
575    AOT_hence φ{α}  χ{α} for α by (metis "→E" "→I")
576    AOT_hence α(φ{α}  χ{α}) using "∀I" by fast
577  }
578  thus ?thesis using "&I" "→I" "&E" by meson
579qed
580
581AOT_theorem "cqt-basic:10": (α(φ{α}  ψ{α}) & α(ψ{α}  χ{α}))  α (φ{α}  χ{α})
582proof(rule "→I"; rule "∀I")
583  fix β
584  AOT_assume α(φ{α}  ψ{α}) & α(ψ{α}  χ{α})
585  AOT_hence φ{β}  ψ{β} and ψ{β}  χ{β} using "&E" "∀E" by blast+
586  AOT_thus φ{β}  χ{β} using "≡I" "≡E" by blast
587qed
588
589AOT_theorem "cqt-basic:11": α(φ{α}  ψ{α})  α (ψ{α}  φ{α})
590proof (rule "≡I"; rule "→I")
591  AOT_assume 0: α(φ{α}  ψ{α})
592  {
593    fix α
594    AOT_have φ{α}  ψ{α} using 0 "∀E" by blast
595    AOT_hence ψ{α}  φ{α} using "≡I" "≡E" "→I" "→E" by metis
596  }
597  AOT_thus α(ψ{α}  φ{α}) using "∀I" by fast
598next
599  AOT_assume 0: α(ψ{α}  φ{α})
600  {
601    fix α
602    AOT_have ψ{α}  φ{α} using 0 "∀E" by blast
603    AOT_hence φ{α}  ψ{α} using "≡I" "≡E" "→I" "→E" by metis
604  }
605  AOT_thus α(φ{α}  ψ{α}) using "∀I" by fast
606qed
607
608AOT_theorem "cqt-basic:12": α φ{α}  α (ψ{α}  φ{α})
609  by (simp add: "∀E"(2) "→I" GEN)
610
611AOT_theorem "cqt-basic:13": α φ{α}  β φ{β}
612  using "≡I" "→I" by blast
613
614AOT_theorem "cqt-basic:14": (α1...∀αn (φ{α1...αn}  ψ{α1...αn}))  ((α1...∀αn φ{α1...αn})  (α1...∀αn ψ{α1...αn}))
615  using "cqt:3"[axiom_inst] by auto
616
617AOT_theorem "cqt-basic:15": (α1...∀αn (φ  ψ{α1...αn}))  (φ  (α1...∀αn ψ{α1...αn}))
618  using "cqt-orig:2" by auto
619
620(* TODO: once more the same in the embedding... need to distinguish these better *)
621AOT_theorem "universal-cor": assumes for arbitrary β: φ{β}  shows α φ{α}
622  using GEN assms .
623
624AOT_theorem "existential:1": assumes φ{τ} and τ shows α φ{α}
625proof(rule "raa-cor:1")
626  AOT_assume ¬α φ{α}
627  AOT_hence α ¬φ{α}
628    using "≡dfI" "conventions:4" RAA "&I" by blast
629  AOT_hence ¬φ{τ} using assms(2) "∀E"(1) "→E" by blast
630  AOT_thus φ{τ} & ¬φ{τ} using assms(1) "&I" by blast
631qed
632
633AOT_theorem "existential:2[const_var]": assumes φ{β} shows α φ{α}
634  using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast
635
636AOT_theorem "existential:2[lambda]":
637  assumes φ{ν1...νn ψ{ν1...νn}]} and INSTANCE_OF_CQT_2(ψ)
638  shows α φ{α}
639  using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
640lemmas "∃I" = "existential:1" "existential:2[const_var]" "existential:2[lambda]" 
641
642AOT_theorem "instantiation":
643  assumes for arbitrary β: φ{β}  ψ and α φ{α}
644  shows ψ
645  by (metis (no_types, lifting) "≡dfE" GEN "raa-cor:3" "conventions:4" assms)
646lemmas "∃E" = "instantiation"
647
648AOT_theorem "cqt-further:1": α φ{α}  α φ{α}
649  using "∀E"(4) "∃I"(2) "→I" by metis
650
651AOT_theorem "cqt-further:2": ¬α φ{α}  α ¬φ{α}
652  using "∀I" "∃I"(2) "→I" RAA by metis
653
654AOT_theorem "cqt-further:3": α φ{α}  ¬α ¬φ{α}
655  using "∀E"(4) "∃E" "→I" RAA
656  by (metis "cqt-further:2" "≡I" "modus-tollens:1")
657
658AOT_theorem "cqt-further:4": ¬α φ{α}  α ¬φ{α}
659  using "∀I" "∃I"(2)"→I" RAA by metis
660
661AOT_theorem "cqt-further:5": α (φ{α} & ψ{α})  (α φ{α} & α ψ{α})
662  by (metis (no_types, lifting) "&E" "&I" "∃E" "∃I"(2) "→I")
663
664AOT_theorem "cqt-further:6": α (φ{α}  ψ{α})  (α φ{α}  α ψ{α})
665  by (metis (mono_tags, lifting) "∃E" "∃I"(2) "∨E"(3) "∨I"(1, 2) "→I" RAA(2))
666
667AOT_theorem "cqt-further:7": α φ{α}  β φ{β} (* TODO: vacuous in the embedding *)
668  by (simp add: "oth-class-taut:3:a")
669
670AOT_theorem "cqt-further:8": (α φ{α} & α ψ{α})  α (φ{α}  ψ{α})
671  by (metis (mono_tags, lifting) "&E" "≡I" "∀E"(2) "→I" GEN)
672
673AOT_theorem "cqt-further:9": (¬α φ{α} & ¬α ψ{α})  α (φ{α}  ψ{α})
674  by (metis (mono_tags, lifting) "&E" "≡I" "∃I"(2) "→I" GEN "raa-cor:4")
675
676AOT_theorem "cqt-further:10": (α φ{α} & ¬α ψ{α})  ¬α (φ{α}  ψ{α})
677proof(rule "→I"; rule "raa-cor:2")
678  AOT_assume 0: α φ{α} & ¬α ψ{α}
679  then AOT_obtain α where φ{α} using "∃E" "&E"(1) by metis
680  moreover AOT_assume α (φ{α}  ψ{α})
681  ultimately AOT_have ψ{α} using "∀E"(4) "≡E"(1) by blast
682  AOT_hence α ψ{α} using "∃I" by blast
683  AOT_thus α ψ{α} & ¬α ψ{α} using 0 "&E"(2) "&I" by blast
684qed
685
686AOT_theorem "cqt-further:11": αβ φ{α,β}  βα φ{α,β}
687  using "≡I" "→I" "∃I"(2) "∃E" by metis
688
689AOT_theorem "log-prop-prop:1":  φ]
690  using "cqt:2[lambda0]"[axiom_inst] by auto
691
692AOT_theorem "log-prop-prop:2": φ
693  by (rule "≡dfI"[OF "existence:3"]) "cqt:2[lambda]"
694
695AOT_theorem "exist-nec": τ  τ
696proof -
697  AOT_have β β
698    by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
699  AOT_thus τ  τ
700    using "cqt:1"[axiom_inst] "→E" by blast
701qed
702
703(* TODO: replace this mechanism by a "proof by types" command *)
704class AOT_Term_id = AOT_Term +
705  assumes "t=t-proper:1"[AOT]: [v  τ = τ'  τ]
706      and "t=t-proper:2"[AOT]: [v  τ = τ'  τ']
707
708instance κ :: AOT_Term_id
709proof
710  AOT_modally_strict {
711    AOT_show κ = κ'  κ for κ κ'
712    proof(rule "→I")
713      AOT_assume κ = κ'
714      AOT_hence O!κ  A!κ
715        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
716           (meson "→I" "∨I"(1) "&E"(1))+
717      AOT_thus κ
718        by (rule "∨E"(1))
719           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
720    qed
721  }
722next
723  AOT_modally_strict {
724    AOT_show κ = κ'  κ' for κ κ'
725    proof(rule "→I")
726      AOT_assume κ = κ'
727      AOT_hence O!κ'  A!κ'
728        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
729           (meson "→I" "∨I" "&E")+
730      AOT_thus κ'
731        by (rule "∨E"(1))
732           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
733    qed
734  }
735qed
736
737instance rel :: (AOT_κs) AOT_Term_id
738proof
739  AOT_modally_strict {
740    AOT_show Π = Π'  Π for Π Π' :: <'a> (* TODO: how to get rid of the fixes? *)
741    proof(rule "→I")
742      AOT_assume Π = Π'
743      AOT_thus Π using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
744    qed
745  }
746next
747  AOT_modally_strict {
748    AOT_show Π = Π'  Π' for Π Π' :: <'a> (* TODO: how to get rid of the fixes? *)
749    proof(rule "→I")
750      AOT_assume Π = Π'
751      AOT_thus Π' using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
752    qed
753  }
754qed
755
756instance 𝗈 :: AOT_Term_id
757proof
758  AOT_modally_strict {
759    fix φ ψ
760    AOT_show φ = ψ  φ
761    proof(rule "→I")
762      AOT_assume φ = ψ
763      AOT_thus φ using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
764    qed
765  }
766next
767  AOT_modally_strict {
768    fix φ ψ
769    AOT_show φ = ψ  ψ
770    proof(rule "→I")
771      AOT_assume φ = ψ
772      AOT_thus ψ using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
773    qed
774  }
775qed
776
777instance prod :: (AOT_Term_id, AOT_Term_id) AOT_Term_id
778proof
779  AOT_modally_strict {
780    fix τ τ' :: 'a×'b
781    AOT_show τ = τ'  τ
782    proof (induct τ; induct τ'; rule "→I")
783      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
784      AOT_assume «(τ1, τ2)» = «(τ1', τ2')»
785      AOT_hence (τ1 = τ1') & (τ2 = τ2') by (metis "≡dfE" tuple_identity_1)
786      AOT_hence τ1 and τ2 using "t=t-proper:1" "&E" "vdash-properties:10" by blast+
787      AOT_thus «(τ1, τ2)» by (metis "≡dfI" "&I" tuple_denotes)
788    qed
789  }
790next
791  AOT_modally_strict {
792    fix τ τ' :: 'a×'b
793    AOT_show τ = τ'  τ'
794    proof (induct τ; induct τ'; rule "→I")
795      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
796      AOT_assume «(τ1, τ2)» = «(τ1', τ2')»
797      AOT_hence (τ1 = τ1') & (τ2 = τ2') by (metis "≡dfE" tuple_identity_1)
798      AOT_hence τ1' and τ2' using "t=t-proper:2" "&E" "vdash-properties:10" by blast+
799      AOT_thus «(τ1', τ2')» by (metis "≡dfI" "&I" tuple_denotes)
800    qed
801  }
802qed
803
804(* TODO: this is the end of the "proof by types" and makes the results available on new theorems *)
805AOT_register_type_constraints
806  Term: _::AOT_Term_id› _::AOT_Term_id›
807AOT_register_type_constraints
808  Individual: ‹κ› _::{AOT_κs, AOT_Term_id}
809AOT_register_type_constraints
810  Relation: <_::{AOT_κs, AOT_Term_id}>
811
812AOT_theorem "id-rel-nec-equiv:1": Π = Π'  x1...∀xn ([Π]x1...xn  [Π']x1...xn)
813proof(rule "→I")
814  AOT_assume assumption: Π = Π'
815  AOT_hence Π and Π'
816    using "t=t-proper:1" "t=t-proper:2" MP by blast+
817  moreover AOT_have FG (F = G  ((x1...∀xn ([F]x1...xn  [F]x1...xn))  x1...∀xn ([F]x1...xn  [G]x1...xn)))
818    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
819  ultimately AOT_have Π = Π'  ((x1...∀xn ([Π]x1...xn  [Π]x1...xn))  x1...∀xn ([Π]x1...xn  [Π']x1...xn))
820    using "∀E"(1) by blast
821  AOT_hence (x1...∀xn ([Π]x1...xn  [Π]x1...xn))  x1...∀xn ([Π]x1...xn  [Π']x1...xn)
822    using assumption "→E" by blast
823  moreover AOT_have x1...∀xn ([Π]x1...xn  [Π]x1...xn)
824    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
825  ultimately AOT_show x1...∀xn ([Π]x1...xn  [Π']x1...xn)
826    using "→E" by blast
827qed
828
829AOT_theorem "id-rel-nec-equiv:2": φ = ψ  (φ  ψ)
830proof(rule "→I")
831  AOT_assume assumption: φ = ψ
832  AOT_hence φ and ψ
833    using "t=t-proper:1" "t=t-proper:2" MP by blast+
834  moreover AOT_have pq (p = q  (((p  p)  (p  q))))
835    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
836  ultimately AOT_have φ = ψ  ((φ  φ)  (φ  ψ))
837    using "∀E"(1) by blast
838  AOT_hence (φ  φ)  (φ  ψ)
839    using assumption "→E" by blast
840  moreover AOT_have (φ  φ)
841    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
842  ultimately AOT_show (φ  ψ)
843    using "→E" by blast
844qed
845
846AOT_theorem "rule=E": assumes φ{τ} and τ = σ shows φ{σ}
847proof -
848  AOT_have τ and σ using assms(2) "t=t-proper:1" "t=t-proper:2" "→E" by blast+
849  moreover AOT_have αβ(α = β  (φ{α}  φ{β}))
850    apply (rule GEN)+ using "l-identity"[axiom_inst] by blast
851  ultimately AOT_have τ = σ  (φ{τ}  φ{σ})
852    using "∀E"(1) by blast
853  AOT_thus φ{σ} using assms "→E" by blast
854qed
855
856AOT_theorem "propositions-lemma:1":  φ] = φ
857proof -
858  AOT_have φ by (simp add: "log-prop-prop:2")
859  moreover AOT_have p  p] = p using "lambda-predicates:3[zero]"[axiom_inst] "∀I" by fast
860  ultimately AOT_show  φ] = φ
861    using "∀E" by blast
862qed
863
864AOT_theorem "propositions-lemma:2":  φ]  φ
865proof -
866  AOT_have  φ]   φ] by (simp add: "oth-class-taut:3:a")
867  AOT_thus  φ]  φ using "propositions-lemma:1" "rule=E" by blast
868qed
869
870(* propositions-lemma:3 through propositions-lemma:5 do not apply *)
871
872AOT_theorem "propositions-lemma:6": (φ  ψ)  ( φ]   ψ])
873  by (metis "≡E"(1) "≡E"(5) "Associativity of ≡" "propositions-lemma:2")
874
875(* dr-alphabetic-rules does not apply *)
876
877AOT_theorem "oa-exist:1": O!
878proof -
879  AOT_have x [E!]x] by "cqt:2[lambda]"
880  AOT_hence 1: O! = x [E!]x] using "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1)] "→E" by blast
881  AOT_show O! using "t=t-proper:1"[THEN "→E", OF 1] by simp
882qed
883
884AOT_theorem "oa-exist:2": A!
885proof -
886  AOT_have x ¬[E!]x] by "cqt:2[lambda]"
887  AOT_hence 1: A! = x ¬[E!]x] using "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1)] "→E" by blast
888  AOT_show A! using "t=t-proper:1"[THEN "→E", OF 1] by simp
889qed
890
891AOT_theorem "oa-exist:3": O!x  A!x
892proof(rule "raa-cor:1")
893  AOT_assume ¬(O!x  A!x)
894  AOT_hence A: ¬O!x and B: ¬A!x
895    using "Disjunction Addition"(1) "modus-tollens:1" "∨I"(2) "raa-cor:5" by blast+
896  AOT_have C: O! = x [E!]x]
897    by (rule "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
898  AOT_have D: A! = x ¬[E!]x]
899    by (rule "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
900  AOT_have E: ¬x [E!]x]x
901    using A C "rule=E" by fast
902  AOT_have F: ¬x ¬[E!]x]x
903    using B D "rule=E" by fast
904  AOT_have G: x [E!]x]x  [E!]x
905    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
906  AOT_have H: x ¬[E!]x]x  ¬[E!]x
907    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
908  AOT_show ¬[E!]x & ¬¬[E!]x using G E "≡E" H F "≡E" "&I" by metis
909qed
910
911AOT_theorem "p-identity-thm2:1": F = G  x(x[F]  x[G])
912proof -
913  AOT_have F = G  F & G & x(x[F]  x[G])
914    using "identity:2" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
915  moreover AOT_have F and G
916    by (auto simp: "cqt:2[const_var]"[axiom_inst])
917  ultimately AOT_show F = G  x(x[F]  x[G])
918    using "≡S"(1) "&I" by blast
919qed
920
921AOT_theorem "p-identity-thm2:2[2]": F = G  y1(x [F]xy1] = x [G]xy1] & x [F]y1x] = x [G]y1x])
922proof -
923  AOT_have F = G  F & G & y1(x [F]xy1] = x [G]xy1] & x [F]y1x] = x [G]y1x])
924    using "identity:3[2]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
925  moreover AOT_have F and G
926    by (auto simp: "cqt:2[const_var]"[axiom_inst])
927  ultimately show ?thesis
928    using "≡S"(1) "&I" by blast
929qed
930    
931AOT_theorem "p-identity-thm2:2[3]": F = G  y1y2(x [F]xy1y2] = x [G]xy1y2] & x [F]y1xy2] = x [G]y1xy2] & x [F]y1y2x] = x [G]y1y2x])
932proof -
933  AOT_have F = G  F & G & y1y2(x [F]xy1y2] = x [G]xy1y2] & x [F]y1xy2] = x [G]y1xy2] & x [F]y1y2x] = x [G]y1y2x])
934    using "identity:3[3]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
935  moreover AOT_have F and G
936    by (auto simp: "cqt:2[const_var]"[axiom_inst])
937  ultimately show ?thesis
938    using "≡S"(1) "&I" by blast
939qed
940
941AOT_theorem "p-identity-thm2:2[4]": F = G  y1y2y3(x [F]xy1y2y3] = x [G]xy1y2y3] & x [F]y1xy2y3] = x [G]y1xy2y3] & x [F]y1y2xy3] = x [G]y1y2xy3] & x [F]y1y2y3x] = x [G]y1y2y3x])
942proof -
943  AOT_have F = G  F & G & y1y2y3(x [F]xy1y2y3] = x [G]xy1y2y3] & x [F]y1xy2y3] = x [G]y1xy2y3] & x [F]y1y2xy3] = x [G]y1y2xy3] & x [F]y1y2y3x] = x [G]y1y2y3x])
944    using "identity:3[4]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
945  moreover AOT_have F and G
946    by (auto simp: "cqt:2[const_var]"[axiom_inst])
947  ultimately show ?thesis
948    using "≡S"(1) "&I" by blast
949qed
950
951AOT_theorem "p-identity-thm2:2":
952  F = G  x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»
953proof -
954  AOT_have F = G  F & G & x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»
955    using "identity:3" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
956  moreover AOT_have F and G
957    by (auto simp: "cqt:2[const_var]"[axiom_inst])
958  ultimately show ?thesis
959    using "≡S"(1) "&I" by blast
960qed
961
962AOT_theorem "p-identity-thm2:3":
963  p = q  x p] = x q]
964proof -
965  AOT_have p = q  p & q & x p] = x q]
966    using "identity:4" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
967  moreover AOT_have p and q
968    by (auto simp: "cqt:2[const_var]"[axiom_inst])
969  ultimately show ?thesis
970    using "≡S"(1) "&I" by blast
971qed
972
973class AOT_Term_id_2 = AOT_Term_id + assumes "id-eq:1": [v  α = α]
974
975instance κ :: AOT_Term_id_2
976proof
977  AOT_modally_strict {
978    fix x
979    {
980      AOT_assume O!x
981      moreover AOT_have F([F]x  [F]x)
982        using RN GEN "oth-class-taut:3:a" by fast
983      ultimately AOT_have O!x & O!x & F([F]x  [F]x) using "&I" by simp
984    }
985    moreover {
986      AOT_assume A!x
987      moreover AOT_have F(x[F]  x[F])
988        using RN GEN "oth-class-taut:3:a" by fast
989      ultimately AOT_have A!x & A!x & F(x[F]  x[F]) using "&I" by simp
990    }
991    ultimately AOT_have (O!x & O!x & F([F]x  [F]x))  (A!x & A!x & F(x[F]  x[F]))
992      using "oa-exist:3" "∨I"(1) "∨I"(2) "∨E"(3) "raa-cor:1" by blast
993    AOT_thus x = x
994      using "identity:1"[THEN "df-rules-formulas[4]"] "→E" by blast
995  }
996qed
997
998instance rel :: ("{AOT_κs,AOT_Term_id_2}") AOT_Term_id_2
999proof
1000  AOT_modally_strict {
1001    fix F :: "<'a> AOT_var"
1002    AOT_have 0: x1...xn [F]x1...xn] = F
1003      by (simp add: "lambda-predicates:3"[axiom_inst])
1004    AOT_have x1...xn [F]x1...xn]
1005      by "cqt:2[lambda]"
1006    AOT_hence x1...xn [F]x1...xn] = x1...xn [F]x1...xn]
1007      using "lambda-predicates:1"[axiom_inst] "→E" by blast
1008    AOT_show F = F using "rule=E" 0 by force 
1009  }
1010qed
1011
1012instance 𝗈 :: AOT_Term_id_2
1013proof
1014  AOT_modally_strict {
1015    fix p
1016    AOT_have 0:  p] = p
1017      by (simp add: "lambda-predicates:3[zero]"[axiom_inst])
1018    AOT_have  p]
1019      by (rule "cqt:2[lambda0]"[axiom_inst])
1020    AOT_hence  p] =  p]
1021      using "lambda-predicates:1[zero]"[axiom_inst] "→E" by blast
1022    AOT_show p = p using "rule=E" 0 by force
1023  }
1024qed
1025
1026instance prod :: (AOT_Term_id_2, AOT_Term_id_2) AOT_Term_id_2
1027proof
1028  AOT_modally_strict {
1029    fix α :: ('a×'b) AOT_var›
1030    AOT_show α = α
1031    proof (induct)
1032      AOT_show τ = τ if τ for τ :: 'a×'b
1033        using that
1034      proof (induct τ)
1035        fix τ1 :: 'a and τ2 :: 'b
1036        AOT_assume «(τ1,τ2)»
1037        AOT_hence τ1 and τ2 using "≡dfE" "&E" tuple_denotes by blast+
1038        AOT_hence τ1 = τ1 and τ2 = τ2 using "id-eq:1"[unvarify α] by blast+
1039        AOT_thus «(τ1, τ2)» = «(τ1, τ2)» by (metis "≡dfI" "&I" tuple_identity_1)
1040      qed
1041    qed
1042  }
1043qed
1044
1045AOT_register_type_constraints
1046  Term: _::AOT_Term_id_2› _::AOT_Term_id_2›
1047AOT_register_type_constraints
1048  Individual: ‹κ› _::{AOT_κs, AOT_Term_id_2}
1049AOT_register_type_constraints
1050  Relation: <_::{AOT_κs, AOT_Term_id_2}>
1051
1052(* TODO: Interestingly, this doesn't depend on "id-eq:1" at all! *)
1053AOT_theorem "id-eq:2": α = β  β = α
1054(*
1055  TODO: look at this proof generated using:
1056        including AOT_no_atp sledgehammer[isar_proofs = true]
1057proof -
1058  have "(∃φ. [v ⊨ ~β = α → ~φ] ∧ [v ⊨ α = β → φ]) ∨ (∃φ. ¬ [v ⊨ φ{α} → φ{β}])"
1059    by meson
1060  then show ?thesis
1061    by (meson "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" l_"identity:1" "useful-tautologies:1")
1062qed
1063*)
1064(*  by (meson "rule=E" "deduction-theorem") *)
1065proof (rule "→I")
1066  AOT_assume α = β
1067  moreover AOT_have β = β using calculation "rule=E"[of _ "λ τ . «τ = β»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1068  moreover AOT_have α = α  α = α using "if-p-then-p" by blast
1069  ultimately AOT_show β = α
1070    using "→E" "→I" "rule=E"[of _ "λ τ . «(τ = τ)  (τ = α)»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1071qed
1072
1073AOT_theorem "id-eq:3": α = β & β = γ  α = γ
1074  using "rule=E" "→I" "&E" by blast
1075
1076AOT_theorem "id-eq:4": α = β  γ (α = γ  β = γ)
1077proof (rule "≡I"; rule "→I")
1078  AOT_assume 0: α = β
1079  AOT_hence 1: β = α using "id-eq:2" "→E" by blast
1080  AOT_show γ (α = γ  β = γ)
1081    by (rule GEN) (metis "≡I" "→I" 0 "1" "rule=E")
1082next
1083  AOT_assume γ (α = γ  β = γ)
1084  AOT_hence α = α  β = α using "∀E"(2) by blast
1085  AOT_hence α = α  β = α using "≡E"(1) "→I" by blast
1086  AOT_hence β = α using "id-eq:1" "→E" by blast
1087  AOT_thus α = β using "id-eq:2" "→E" by blast
1088qed
1089
1090AOT_theorem "rule=I:1": assumes τ shows τ = τ
1091proof -
1092  AOT_have α (α = α)
1093    by (rule GEN) (metis "id-eq:1")
1094  AOT_thus τ = τ using assms "∀E" by blast
1095qed
1096
1097AOT_theorem "rule=I:2[const_var]": "α = α"
1098  using "id-eq:1".
1099
1100AOT_theorem "rule=I:2[lambda]": assumes INSTANCE_OF_CQT_2(φ) shows "ν1...νn φ{ν1...νn}] = ν1...νn φ{ν1...νn}]"
1101proof -
1102  AOT_have α (α = α)
1103    by (rule GEN) (metis "id-eq:1")
1104  moreover AOT_have ν1...νn φ{ν1...νn}] using assms by (rule "cqt:2[lambda]"[axiom_inst])
1105  ultimately AOT_show ν1...νn φ{ν1...νn}] = ν1...νn φ{ν1...νn}] using assms "∀E" by blast
1106qed
1107
1108lemmas "=I" = "rule=I:1" "rule=I:2[const_var]" "rule=I:2[lambda]"
1109
1110AOT_theorem "rule-id-df:1":
1111  assumes τ{α1...αn} =df σ{α1...αn} and σ{τ1...τn}
1112  shows τ{τ1...τn} = σ{τ1...τn}
1113proof -
1114  AOT_have σ{τ1...τn}  τ{τ1...τn} = σ{τ1...τn}
1115    using "df-rules-terms[3]" assms(1) "&E" by blast
1116  AOT_thus τ{τ1...τn} = σ{τ1...τn}
1117    using assms(2) "→E" by blast
1118qed
1119
1120AOT_theorem "rule-id-df:1[zero]":
1121  assumes τ =df σ and σ
1122  shows τ = σ
1123proof -
1124  AOT_have σ  τ = σ
1125    using "df-rules-terms[4]" assms(1) "&E" by blast
1126  AOT_thus τ = σ
1127    using assms(2) "→E" by blast
1128qed
1129
1130AOT_theorem "rule-id-df:2:a":
1131  assumes τ{α1...αn} =df σ{α1...αn} and σ{τ1...τn} and φ{τ{τ1...τn}}
1132  shows φ{σ{τ1...τn}}
1133proof -
1134  AOT_have τ{τ1...τn} = σ{τ1...τn} using "rule-id-df:1" assms(1,2) by blast
1135  AOT_thus φ{σ{τ1...τn}} using assms(3) "rule=E" by blast
1136qed
1137
1138(* TODO: get rid of this, ideally *)
1139AOT_theorem "rule-id-df:2:a[2]":
1140  assumes τ{«(α1,α2)»} =df σ{«(α1,α2)»} and σ{«(τ1,τ2)»} and φ{τ{«(τ1,τ2)»}}
1141  shows φ{σ{«(τ1,τ2)»}}
1142proof -
1143  AOT_have τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}
1144  proof -
1145    AOT_have σ{«(τ1,τ2)»}  τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}
1146      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1147    AOT_thus τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}
1148      using assms(2) "→E" by blast
1149  qed
1150  AOT_thus φ{σ{«(τ1,τ2)»}} using assms(3) "rule=E" by blast
1151qed
1152
1153AOT_theorem "rule-id-df:2:a[zero]":
1154  assumes τ =df σ and σ and φ{τ}
1155  shows φ{σ}
1156proof -
1157  AOT_have τ = σ using "rule-id-df:1[zero]" assms(1,2) by blast
1158  AOT_thus φ{σ} using assms(3) "rule=E" by blast
1159qed
1160
1161lemmas "=dfE" = "rule-id-df:2:a" "rule-id-df:2:a[zero]"
1162
1163AOT_theorem "rule-id-df:2:b":
1164  assumes τ{α1...αn} =df σ{α1...αn} and σ{τ1...τn} and φ{σ{τ1...τn}}
1165  shows φ{τ{τ1...τn}}
1166proof -
1167  AOT_have τ{τ1...τn} = σ{τ1...τn} using "rule-id-df:1" assms(1,2) by blast
1168  AOT_hence σ{τ1...τn} = τ{τ1...τn}
1169    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1170  AOT_thus φ{τ{τ1...τn}} using assms(3) "rule=E" by blast
1171qed
1172
1173(* TODO: get rid of this, ideally *)
1174AOT_theorem "rule-id-df:2:b[2]":
1175  assumes τ{«(α1,α2)»} =df σ{«(α1,α2)»} and σ{«(τ1,τ2)»} and φ{σ{«(τ1,τ2)»}}
1176  shows φ{τ{«(τ1,τ2)»}}
1177proof -
1178  AOT_have τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}
1179  proof -
1180    AOT_have σ{«(τ1,τ2)»}  τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}
1181      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1182    AOT_thus τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}
1183      using assms(2) "→E" by blast
1184  qed
1185  AOT_hence σ{«(τ1,τ2)»} = τ{«(τ1,τ2)»}
1186    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1187  AOT_thus φ{τ{«(τ1,τ2)»}} using assms(3) "rule=E" by blast
1188qed
1189
1190AOT_theorem "rule-id-df:2:b[zero]":
1191  assumes τ =df σ and σ and φ{σ}
1192  shows φ{τ}
1193proof -
1194  AOT_have τ = σ using "rule-id-df:1[zero]" assms(1,2) by blast
1195  AOT_hence σ = τ
1196    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1197  AOT_thus φ{τ} using assms(3) "rule=E" by blast
1198qed
1199
1200lemmas "=dfI" = "rule-id-df:2:b" "rule-id-df:2:b[zero]"
1201
1202AOT_theorem "free-thms:1": τ  β (β = τ)
1203  by (metis "∃E" "rule=I:1" "t=t-proper:2" "→I" "∃I"(1) "≡I" "→E")
1204
1205AOT_theorem "free-thms:2": α φ{α}  (β (β = τ)  φ{τ})
1206  by (metis "∃E" "rule=E" "cqt:2[const_var]"[axiom_inst] "→I" "∀E"(1))
1207
1208AOT_theorem "free-thms:3[const_var]": β (β = α)
1209  by (meson "∃I"(2) "id-eq:1")
1210
1211AOT_theorem "free-thms:3[lambda]": assumes INSTANCE_OF_CQT_2(φ) shows β (β = ν1...νn φ{ν1...νn}])
1212  by (meson "=I"(3) assms "cqt:2[lambda]"[axiom_inst] "existential:1")
1213
1214AOT_theorem "free-thms:4[rel]": ([Π]κ1...κn  κ1...κn[Π])  β (β = Π)
1215  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1216
1217(* TODO: this is a rather weird way to formulate this and we don't have tuple-existential-elimination
1218         or tuple-equality-elimination in the theory... Splitting them is also a bit unfortunate, though.*)
1219AOT_theorem "free-thms:4[vars]": ([Π]κ1...κn  κ1...κn[Π])  β1...∃βn (β1...βn = κ1...κn)
1220  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1221
1222AOT_theorem "free-thms:4[1,rel]": ([Π]κ  κ[Π])  β (β = Π)
1223  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1224AOT_theorem "free-thms:4[1,1]": ([Π]κ  κ[Π])  β (β = κ)
1225  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1226
1227AOT_theorem "free-thms:4[2,rel]": ([Π]κ1κ2  κ1κ2[Π])  β (β = Π)
1228  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1229AOT_theorem "free-thms:4[2,1]": ([Π]κ1κ2  κ1κ2[Π])  β (β = κ1)
1230  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1231AOT_theorem "free-thms:4[2,2]": ([Π]κ1κ2  κ1κ2[Π])  β (β = κ2)
1232  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1233AOT_theorem "free-thms:4[3,rel]": ([Π]κ1κ2κ3  κ1κ2κ3[Π])  β (β = Π)
1234  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1235AOT_theorem "free-thms:4[3,1]": ([Π]κ1κ2κ3  κ1κ2κ3[Π])  β (β = κ1)
1236  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1237AOT_theorem "free-thms:4[3,2]": ([Π]κ1κ2κ3  κ1κ2κ3[Π])  β (β = κ2)
1238  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1239AOT_theorem "free-thms:4[3,3]": ([Π]κ1κ2κ3  κ1κ2κ3[Π])  β (β = κ3)
1240  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1241AOT_theorem "free-thms:4[4,rel]": ([Π]κ1κ2κ3κ4  κ1κ2κ3κ4[Π])  β (β = Π)
1242  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1243AOT_theorem "free-thms:4[4,1]": ([Π]κ1κ2κ3κ4  κ1κ2κ3κ4[Π])  β (β = κ1)
1244  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1245AOT_theorem "free-thms:4[4,2]": ([Π]κ1κ2κ3κ4  κ1κ2κ3κ4[Π])  β (β = κ2)
1246  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1247AOT_theorem "free-thms:4[4,3]": ([Π]κ1κ2κ3κ4  κ1κ2κ3κ4[Π])  β (β = κ3)
1248  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1249AOT_theorem "free-thms:4[4,4]": ([Π]κ1κ2κ3κ4  κ1κ2κ3κ4[Π])  β (β = κ4)
1250  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1251
1252AOT_theorem "ex:1:a": α α
1253  by (rule GEN) (fact "cqt:2[const_var]"[axiom_inst])
1254AOT_theorem "ex:1:b": αβ(β = α)
1255  by (rule GEN) (fact "free-thms:3[const_var]")
1256
1257AOT_theorem "ex:2:a": α
1258  by (rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1259AOT_theorem "ex:2:b": β(β = α)
1260  by (rule RN) (fact "free-thms:3[const_var]")
1261
1262AOT_theorem "ex:3:a": α α
1263  by (rule RN) (fact "ex:1:a")
1264AOT_theorem "ex:3:b": αβ(β = α)
1265  by (rule RN) (fact "ex:1:b")
1266
1267AOT_theorem "ex:4:a": α α
1268  by (rule GEN; rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1269AOT_theorem "ex:4:b": αβ(β = α)
1270  by (rule GEN; rule RN) (fact "free-thms:3[const_var]")
1271
1272AOT_theorem "ex:5:a": α α
1273  by (rule RN) (simp add: "ex:4:a")
1274AOT_theorem "ex:5:b": αβ(β = α)
1275  by (rule RN) (simp add: "ex:4:b")
1276
1277AOT_theorem "all-self=:1": α(α = α)
1278  by (rule RN; rule GEN) (fact "id-eq:1")
1279AOT_theorem "all-self=:2": α(α = α)
1280  by (rule GEN; rule RN) (fact "id-eq:1")
1281
1282AOT_theorem "id-nec:1": α = β  (α = β)
1283proof(rule "→I")
1284  AOT_assume α = β
1285  moreover AOT_have (α = α)
1286    by (rule RN) (fact "id-eq:1")
1287  ultimately AOT_show (α = β) using "rule=E" by fast
1288qed
1289
1290AOT_theorem "id-nec:2": τ = σ  (τ = σ)
1291proof(rule "→I")
1292  AOT_assume asm: τ = σ
1293  moreover AOT_have τ
1294    using calculation "t=t-proper:1" "→E" by blast
1295  moreover AOT_have (τ = τ)
1296    using calculation "all-self=:2" "∀E"(1) by blast
1297  ultimately AOT_show (τ = σ) using "rule=E" by fast
1298qed
1299
1300AOT_theorem "term-out:1": φ{α}  β (β = α & φ{β})
1301proof (rule "≡I"; rule "→I")
1302  AOT_assume asm: φ{α}
1303  AOT_show β (β = α & φ{β})
1304    by (rule "∃I"(2)[where β=α]; rule "&I")
1305       (auto simp: "id-eq:1" asm)
1306next
1307  AOT_assume 0: β (β = α & φ{β})
1308  (* TODO: have another look at this instantiation. Ideally AOT_obtain would resolve directly to the existential
1309           statement as proof obligation *)
1310  AOT_obtain β where β = α & φ{β} using "instantiation"[rotated, OF 0] by blast
1311  AOT_thus φ{α} using "&E" "rule=E" by blast
1312qed
1313
1314AOT_theorem "term-out:2": τ  (φ{τ}  α(α = τ & φ{α}))
1315proof(rule "→I")
1316  AOT_assume τ
1317  moreover AOT_have α (φ{α}  β (β = α & φ{β}))
1318    by (rule GEN) (fact "term-out:1")
1319  ultimately AOT_show φ{τ}  α(α = τ & φ{α})
1320    using "∀E" by blast
1321qed
1322
1323(* TODO: example of an apply-style proof. Keep or reformulate? *)
1324AOT_theorem "term-out:3": (φ{α} & β(φ{β}  β = α))  β(φ{β}  β = α)
1325  apply (rule "≡I"; rule "→I")
1326   apply (frule "&E"(1)) apply (drule "&E"(2))
1327   apply (rule GEN; rule "≡I"; rule "→I")
1328  using "rule-ui:2[const_var]" "vdash-properties:5" apply blast
1329  apply (meson "rule=E" "id-eq:1")
1330  apply (rule "&I")
1331  using "id-eq:1" "≡E"(2) "rule-ui:3" apply blast
1332  apply (rule GEN; rule "→I")
1333  using "≡E"(1) "rule-ui:2[const_var]" by blast
1334
1335AOT_theorem "term-out:4": (φ{β} & α(φ{α}  α = β))  α(φ{α}  α = β)
1336  using "term-out:3" . (* TODO: same as above - another instance of the generalized alphabetic variant rule... *)
1337
1338(* TODO: would of course be nice to define it without the syntax magic *)
1339AOT_define AOT_exists_unique :: ‹α  φ  φ›
1340  "uniqueness:1": «AOT_exists_unique φ» df α (φ{α} & β (φ{β}  β = α))
1341syntax "_AOT_exists_unique" :: ‹α  φ  φ› ("∃!_ _" [1,40])
1342AOT_syntax_print_translations
1343  "_AOT_exists_unique τ φ" <= "CONST AOT_exists_unique (_abs τ φ)"
1344syntax
1345   "_AOT_exists_unique_ellipse" :: ‹id_position  id_position  φ  φ› (∃!_...∃!_ _› [1,40])
1346parse_ast_translation[(syntax_const‹_AOT_exists_unique_ellipse›, fn ctx => fn [a,b,c] =>
1347  Ast.mk_appl (Ast.Constant "AOT_exists_unique") [parseEllipseList "_AOT_vars" ctx [a,b],c]),
1348(syntax_const‹_AOT_exists_unique›,AOT_restricted_binder const_name‹AOT_exists_unique› const_syntax‹AOT_conj›)]
1349print_translationAOT_syntax_print_translations
1350  [AOT_preserve_binder_abs_tr' const_syntax‹AOT_exists_unique› syntax_const‹_AOT_exists_unique› (syntax_const‹_AOT_exists_unique_ellipse›, true) const_name‹AOT_conj›,
1351  AOT_binder_trans @{theory} @{binding "AOT_exists_unique_binder"} syntax_const‹_AOT_exists_unique›]
1352
1353
1354
1355context AOT_meta_syntax
1356begin
1357notation AOT_exists_unique (binder "!" 20)
1358end
1359context AOT_no_meta_syntax
1360begin
1361no_notation AOT_exists_unique (binder "!" 20)
1362end
1363
1364AOT_theorem "uniqueness:2": ∃!α φ{α}  αβ(φ{β}  β = α)
1365proof(rule "≡I"; rule "→I")
1366    AOT_assume ∃!α φ{α}
1367    AOT_hence α (φ{α} & β (φ{β}  β = α))
1368      using "uniqueness:1" "≡dfE" by blast
1369    then AOT_obtain α where φ{α} & β (φ{β}  β = α) using "instantiation"[rotated] by blast
1370    AOT_hence β(φ{β}  β = α) using "term-out:3" "≡E" by blast
1371    AOT_thus αβ(φ{β}  β = α)
1372      using "∃I" by fast
1373next
1374    AOT_assume αβ(φ{β}  β = α)
1375    then AOT_obtain α where β (φ{β}  β = α) using "instantiation"[rotated] by blast
1376    AOT_hence φ{α} & β (φ{β}  β = α) using "term-out:3" "≡E" by blast
1377    AOT_hence α (φ{α} & β (φ{β}  β = α))
1378      using "∃I" by fast
1379    AOT_thus ∃!α φ{α}
1380      using "uniqueness:1" "≡dfI" by blast
1381qed
1382
1383AOT_theorem "uni-most": ∃!α φ{α}  βγ((φ{β} & φ{γ})  β = γ)
1384proof(rule "→I"; rule GEN; rule GEN; rule "→I")
1385  fix β γ
1386  AOT_assume ∃!α φ{α}
1387  AOT_hence αβ(φ{β}  β = α)
1388    using "uniqueness:2" "≡E" by blast
1389  then AOT_obtain α where β(φ{β}  β = α)
1390    using "instantiation"[rotated] by blast
1391  moreover AOT_assume φ{β} & φ{γ}
1392  ultimately AOT_have β = α and γ = α
1393    using "∀E"(2) "&E" "≡E"(1,2) by blast+
1394  AOT_thus β = γ
1395    by (metis "rule=E" "id-eq:2" "→E")
1396qed
1397
1398AOT_theorem "nec-exist-!": α(φ{α}  φ{α})  (∃!α φ{α}  ∃!α φ{α})
1399proof (rule "→I"; rule "→I")
1400  AOT_assume a: α(φ{α}  φ{α})
1401  AOT_assume ∃!α φ{α}
1402  AOT_hence α (φ{α} & β (φ{β}  β = α)) using "uniqueness:1" "≡dfE" by blast
1403  then AOT_obtain α where ξ: φ{α} & β (φ{β}  β = α) using "instantiation"[rotated] by blast
1404  AOT_have φ{α}
1405    using ξ a "&E" "∀E" "→E" by fast
1406  moreover AOT_have β (φ{β}  β = α)
1407    apply (rule GEN; rule "→I")
1408    using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
1409  ultimately AOT_have (φ{α} & β (φ{β}  β = α))
1410    using "&I" by blast
1411  AOT_thus ∃!α φ{α}
1412    using "uniqueness:1" "≡dfI" "∃I" by fast
1413qed
1414
1415AOT_theorem "act-cond": 𝒜(φ  ψ)  (𝒜φ  𝒜ψ)
1416  using "→I" "≡E"(1) "logic-actual-nec:2"[axiom_inst] by blast
1417
1418AOT_theorem "nec-imp-act": φ  𝒜φ
1419  by (metis "act-cond" "contraposition:1[2]" "≡E"(4) "qml:2"[THEN act_closure, axiom_inst] "qml-act:2"[axiom_inst] RAA(1) "→E" "→I")
1420
1421AOT_theorem "act-conj-act:1": 𝒜(𝒜φ  φ)
1422  using "→I" "≡E"(2) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] by blast
1423
1424AOT_theorem "act-conj-act:2": 𝒜(φ  𝒜φ)
1425  by (metis "→I" "≡E"(2, 4) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] RAA(1))
1426
1427AOT_theorem "act-conj-act:3": (𝒜φ & 𝒜ψ)  𝒜(φ & ψ)
1428proof -
1429  AOT_have (φ  (ψ  (φ & ψ)))
1430    by (rule RN) (fact Adjunction)
1431  AOT_hence 𝒜(φ  (ψ  (φ & ψ)))
1432    using "nec-imp-act" "→E" by blast
1433  AOT_hence 𝒜φ  𝒜(ψ  (φ & ψ))
1434    using "act-cond" "→E" by blast
1435  moreover AOT_have 𝒜(ψ  (φ & ψ))  (𝒜ψ  𝒜(φ & ψ))
1436    by (fact "act-cond")
1437  ultimately AOT_have 𝒜φ  (𝒜ψ  𝒜(φ & ψ))
1438    using "→I" "→E" by metis
1439  AOT_thus (𝒜φ & 𝒜ψ)  𝒜(φ & ψ)
1440    by (metis Importation "→E")
1441qed
1442
1443AOT_theorem "act-conj-act:4": 𝒜(𝒜φ  φ)
1444proof -
1445  AOT_have (𝒜(𝒜φ  φ) & 𝒜(φ  𝒜φ))  𝒜((𝒜φ  φ) & (φ  𝒜φ))
1446    by (fact "act-conj-act:3")
1447  moreover AOT_have 𝒜(𝒜φ  φ) & 𝒜(φ  𝒜φ)
1448    using "&I" "act-conj-act:1" "act-conj-act:2" by simp
1449  ultimately AOT_have ζ: 𝒜((𝒜φ  φ) & (φ  𝒜φ))
1450    using "→E" by blast
1451  AOT_have 𝒜(((𝒜φ  φ) & (φ  𝒜φ))  (𝒜φ  φ))
1452    using "conventions:3"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst] by blast
1453  AOT_hence 𝒜((𝒜φ  φ) & (φ  𝒜φ))  𝒜(𝒜φ  φ)
1454    using "act-cond" "→E" by blast
1455  AOT_thus 𝒜(𝒜φ  φ) using ζ "→E" by blast
1456qed
1457
1458(* TODO: consider introducing AOT_inductive *)
1459inductive arbitrary_actualization for φ where
1460  arbitrary_actualization φ «𝒜φ»
1461| arbitrary_actualization φ «𝒜ψ» if arbitrary_actualization φ ψ
1462declare arbitrary_actualization.cases[AOT] arbitrary_actualization.induct[AOT]
1463        arbitrary_actualization.simps[AOT] arbitrary_actualization.intros[AOT]
1464syntax arbitrary_actualization :: ‹φ'  φ'  AOT_prop› ("ARBITRARY'_ACTUALIZATION'(_,_')")
1465
1466notepad
1467begin
1468  AOT_modally_strict {
1469    fix φ
1470    AOT_have ARBITRARY_ACTUALIZATION(𝒜φ  φ, 𝒜(𝒜φ  φ))
1471      using AOT_PLM.arbitrary_actualization.intros by metis
1472    AOT_have ARBITRARY_ACTUALIZATION(𝒜φ  φ, 𝒜𝒜(𝒜φ  φ))
1473      using AOT_PLM.arbitrary_actualization.intros by metis
1474    AOT_have ARBITRARY_ACTUALIZATION(𝒜φ  φ, 𝒜𝒜𝒜(𝒜φ  φ))
1475      using AOT_PLM.arbitrary_actualization.intros by metis
1476  }
1477end
1478
1479
1480AOT_theorem "closure-act:1": assumes ARBITRARY_ACTUALIZATION(𝒜φ  φ, ψ) shows ψ
1481using assms proof(induct)
1482  case 1
1483  AOT_show 𝒜(𝒜φ  φ)
1484    by (simp add: "act-conj-act:4")
1485next
1486  case (2 ψ)
1487  AOT_thus 𝒜ψ
1488    by (metis arbitrary_actualization.simps "≡E"(1) "logic-actual-nec:4"[axiom_inst])
1489qed
1490
1491AOT_theorem "closure-act:2": α 𝒜(𝒜φ{α}  φ{α})
1492  by (simp add: "act-conj-act:4" "∀I")
1493
1494AOT_theorem "closure-act:3": 𝒜α 𝒜(𝒜φ{α}  φ{α})
1495  by (metis (no_types, lifting) "act-conj-act:4" "≡E"(1,2) "logic-actual-nec:3"[axiom_inst] "logic-actual-nec:4"[axiom_inst] "∀I")
1496
1497AOT_theorem "closure-act:4": 𝒜α1...∀αn 𝒜(𝒜φ{α1...αn}  φ{α1...αn})
1498  using "closure-act:3" .
1499
1500(* TODO: examine these proofs *)
1501AOT_theorem "RA[1]": assumes  φ shows  𝒜φ
1502  (* This proof is the one rejected in remark (136) (meta-rule) *)
1503  using "¬¬E" assms "≡E"(3) "logic-actual"[act_axiom_inst] "logic-actual-nec:1"[axiom_inst] "modus-tollens:2" by blast
1504AOT_theorem "RA[2]": assumes  φ shows 𝒜φ
1505  (* This is actually Γ ⊢ φ ⟹ □Γ ⊢ 𝒜φ*)
1506  using RN assms "nec-imp-act" "vdash-properties:5" by blast
1507AOT_theorem "RA[3]": assumes Γ  φ shows 𝒜Γ  𝒜φ
1508  using assms by (meson AOT_sem_act imageI)
1509  (* This is not exactly right either. *)
1510
1511AOT_act_theorem "ANeg:1": ¬𝒜φ  ¬φ
1512  by (simp add: "RA[1]" "contraposition:1[1]" "deduction-theorem" "≡I" "logic-actual"[act_axiom_inst])
1513
1514AOT_act_theorem "ANeg:2": ¬𝒜¬φ  φ
1515  using "ANeg:1" "≡I" "≡E"(5) "useful-tautologies:1" "useful-tautologies:2" by blast
1516
1517AOT_theorem "Act-Basic:1": 𝒜φ  𝒜¬φ
1518  by (meson "∨I"(1,2) "≡E"(2) "logic-actual-nec:1"[axiom_inst] "raa-cor:1")
1519
1520AOT_theorem "Act-Basic:2": 𝒜(φ & ψ)  (𝒜φ & 𝒜ψ)
1521proof (rule "≡I"; rule "→I")
1522  AOT_assume 𝒜(φ & ψ)
1523  moreover AOT_have 𝒜((φ & ψ)  φ)
1524    by (simp add: "RA[2]" "Conjunction Simplification"(1))
1525  moreover AOT_have 𝒜((φ & ψ)  ψ)
1526    by (simp add: "RA[2]" "Conjunction Simplification"(2))
1527  ultimately AOT_show 𝒜φ & 𝒜ψ
1528    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1529next
1530  AOT_assume 𝒜φ & 𝒜ψ
1531  AOT_thus 𝒜(φ & ψ)
1532    using "act-conj-act:3" "vdash-properties:6" by blast
1533qed
1534
1535AOT_theorem "Act-Basic:3": 𝒜(φ  ψ)  (𝒜(φ  ψ) & 𝒜(ψ  φ))
1536proof (rule "≡I"; rule "→I")
1537  AOT_assume 𝒜(φ  ψ)
1538  moreover AOT_have 𝒜((φ  ψ)  (φ  ψ))
1539    by (simp add: "RA[2]" "deduction-theorem" "≡E"(1))
1540  moreover AOT_have 𝒜((φ  ψ)  (ψ  φ))
1541    by (simp add: "RA[2]" "deduction-theorem" "≡E"(2))
1542  ultimately AOT_show 𝒜(φ  ψ) & 𝒜(ψ  φ)
1543    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1544next
1545  AOT_assume 𝒜(φ  ψ) & 𝒜(ψ  φ)
1546  AOT_hence 𝒜((φ  ψ) & (ψ  φ))
1547    by (metis "act-conj-act:3" "vdash-properties:10")
1548  moreover AOT_have 𝒜(((φ  ψ) & (ψ  φ))  (φ  ψ))
1549    by (simp add: "conventions:3" "RA[2]" "df-rules-formulas[2]" "vdash-properties:1[2]")
1550  ultimately AOT_show 𝒜(φ  ψ)
1551    using "act-cond"[THEN "→E", THEN "→E"] by metis
1552qed
1553
1554AOT_theorem "Act-Basic:4": (𝒜(φ  ψ) & 𝒜(ψ  φ))  (𝒜φ  𝒜ψ)
1555proof (rule "≡I"; rule "→I")
1556  AOT_assume 0: 𝒜(φ  ψ) & 𝒜(ψ  φ)
1557  AOT_show 𝒜φ  𝒜ψ
1558    using 0 "&E" "act-cond"[THEN "→E", THEN "→E"] "≡I" "→I" by metis
1559next
1560  AOT_assume 𝒜φ  𝒜ψ
1561  AOT_thus 𝒜(φ  ψ) & 𝒜(ψ  φ)
1562    by (metis "→I" "logic-actual-nec:2"[axiom_inst] "≡E"(1,2) "&I")
1563qed
1564
1565AOT_theorem "Act-Basic:5": 𝒜(φ  ψ)  (𝒜φ  𝒜ψ)
1566  using "Act-Basic:3" "Act-Basic:4" "≡E"(5) by blast
1567
1568AOT_theorem "Act-Basic:6": 𝒜φ  𝒜φ
1569  by (simp add: "≡I" "qml:2"[axiom_inst] "qml-act:1"[axiom_inst])
1570
1571AOT_theorem "Act-Basic:7": 𝒜φ  𝒜φ
1572  by (metis "Act-Basic:6" "→I" "→E" "≡E"(1,2) "nec-imp-act" "qml-act:2"[axiom_inst])
1573
1574AOT_theorem "Act-Basic:8": φ  𝒜φ
1575  using "Hypothetical Syllogism" "nec-imp-act" "qml-act:1"[axiom_inst] by blast
1576
1577AOT_theorem "Act-Basic:9": 𝒜(φ  ψ)  (𝒜φ  𝒜ψ)
1578proof (rule "≡I"; rule "→I")
1579  AOT_assume 𝒜(φ  ψ)
1580  AOT_thus 𝒜φ  𝒜ψ
1581  proof (rule "raa-cor:3")
1582    AOT_assume ¬(𝒜φ  𝒜ψ)
1583    AOT_hence ¬𝒜φ & ¬𝒜ψ
1584      by (metis "≡E"(1) "oth-class-taut:5:d")
1585    AOT_hence 𝒜¬φ & 𝒜¬ψ
1586      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] "&E" "&I" by metis
1587    AOT_hence 𝒜(¬φ & ¬ψ)
1588      using "≡E" "Act-Basic:2" by metis
1589    moreover AOT_have 𝒜((¬φ & ¬ψ)  ¬(φ  ψ))
1590      using "RA[2]" "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:d" by blast
1591    moreover AOT_have 𝒜(¬φ & ¬ψ)  𝒜(¬(φ  ψ))
1592      using calculation(2) by (metis "Act-Basic:5" "≡E"(1))
1593    ultimately AOT_have 𝒜(¬(φ  ψ)) using "≡E" by blast
1594    AOT_thus ¬𝒜(φ  ψ)
1595      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by auto
1596  qed
1597next
1598  AOT_assume 𝒜φ  𝒜ψ
1599  AOT_thus 𝒜(φ  ψ)
1600    by (meson "RA[2]" "act-cond" "∨I"(1) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
1601qed
1602
1603AOT_theorem "Act-Basic:10": 𝒜α φ{α}  α 𝒜φ{α}
1604proof -
1605  AOT_have θ: ¬𝒜α ¬φ{α}  ¬α 𝒜¬φ{α}
1606    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1607       (metis "logic-actual-nec:3"[axiom_inst])
1608  AOT_have ξ: ¬α 𝒜¬φ{α}  ¬α ¬𝒜φ{α}
1609    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1610       (rule "logic-actual-nec:1"[THEN universal_closure, axiom_inst, THEN "cqt-basic:3"[THEN "→E"]])
1611  AOT_have 𝒜(α φ{α})  𝒜(¬α ¬φ{α})
1612    using "conventions:4"[THEN "df-rules-formulas[1]", THEN act_closure, axiom_inst]
1613          "conventions:4"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst]
1614    "Act-Basic:4"[THEN "≡E"(1)] "&I" "Act-Basic:5"[THEN "≡E"(2)] by metis
1615  also AOT_have   ¬𝒜α ¬φ{α}
1616    by (simp add: "logic-actual-nec:1" "vdash-properties:1[2]")
1617  also AOT_have   ¬α 𝒜 ¬φ{α} using θ by blast
1618  also AOT_have   ¬α ¬𝒜 φ{α} using ξ by blast
1619  also AOT_have   α 𝒜 φ{α}
1620    using "conventions:4"[THEN "≡Df"] by (metis "≡E"(6) "oth-class-taut:3:a")
1621  finally AOT_show 𝒜α φ{α}  α 𝒜φ{α} .
1622qed
1623
1624
1625AOT_theorem "Act-Basic:11": 𝒜α(φ{α}  ψ{α})  α(𝒜φ{α}  𝒜ψ{α})
1626proof(rule "≡I"; rule "→I")
1627  AOT_assume 𝒜α(φ{α}  ψ{α})
1628  AOT_hence α𝒜(φ{α}  ψ{α})
1629    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] by blast
1630  AOT_hence 𝒜(φ{α}  ψ{α}) for α using "∀E" by blast
1631  AOT_hence 𝒜φ{α}  𝒜ψ{α} for α by (metis "Act-Basic:5" "≡E"(1))
1632  AOT_thus α(𝒜φ{α}  𝒜ψ{α}) by (rule "∀I")
1633next
1634  AOT_assume α(𝒜φ{α}  𝒜ψ{α})
1635  AOT_hence 𝒜φ{α}  𝒜ψ{α} for α using "∀E" by blast
1636  AOT_hence 𝒜(φ{α}  ψ{α}) for α by (metis "Act-Basic:5" "≡E"(2))
1637  AOT_hence α 𝒜(φ{α}  ψ{α}) by (rule "∀I")
1638  AOT_thus 𝒜α(φ{α}  ψ{α})
1639    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
1640qed
1641
1642AOT_act_theorem "act-quant-uniq": β(𝒜φ{β}  β = α)  β(φ{β}  β = α)
1643proof(rule "≡I"; rule "→I")
1644  AOT_assume β(𝒜φ{β}  β = α)
1645  AOT_hence 𝒜φ{β}  β = α for β using "∀E" by blast
1646  AOT_hence φ{β}  β = α for β
1647    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1648    by metis
1649  AOT_thus β(φ{β}  β = α) by (rule "∀I")
1650next
1651  AOT_assume β(φ{β}  β = α)
1652  AOT_hence φ{β}  β = α for β using "∀E" by blast
1653  AOT_hence 𝒜φ{β}  β = α for β
1654    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1655    by metis
1656  AOT_thus β(𝒜φ{β}  β = α) by (rule "∀I")
1657qed
1658
1659AOT_act_theorem "fund-cont-desc": x = ιx(φ{x})  z(φ{z}  z = x)
1660  using descriptions[axiom_inst] "act-quant-uniq" "≡E"(5) by fast
1661
1662AOT_act_theorem hintikka: x = ιx(φ{x})  (φ{x} & z (φ{z}  z = x))
1663  using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" "fund-cont-desc" "≡E"(5) by blast
1664
1665
1666locale russel_axiom =
1667  fixes ψ
1668  assumes ψ_denotes_asm: "[v  ψ{κ}]  [v  κ]"
1669begin
1670AOT_act_theorem "russell-axiom": ψ{ιx φ{x}}  x(φ{x} & z(φ{z}  z = x) & ψ{x})
1671proof -
1672  AOT_have b: x (x = ιx φ{x}  (φ{x} & z(φ{z}  z = x)))
1673    using hintikka "∀I" by fast
1674  show ?thesis
1675  proof(rule "≡I"; rule "→I")
1676    AOT_assume c: ψ{ιx φ{x}}
1677    AOT_hence d: ιx φ{x} using ψ_denotes_asm by blast
1678    AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1679    then AOT_obtain a where a_def: a = ιx φ{x} using "instantiation"[rotated] by blast
1680    moreover AOT_have a = ιx φ{x}  (φ{a} & z(φ{z}  z = a)) using b "∀E" by blast
1681    ultimately AOT_have φ{a} & z(φ{z}  z = a) using "≡E" by blast
1682    moreover AOT_have ψ{a}
1683    proof - 
1684      AOT_have 1: xy(x = y  y = x)
1685        by (simp add: "id-eq:2" "universal-cor")
1686      AOT_have a = ιx φ{x}   ιx φ{x} = a
1687        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1688           (auto simp: 1 d "universal-cor")
1689      AOT_thus ψ{a}
1690        using a_def c "rule=E" "→E" by blast
1691    qed
1692    ultimately AOT_have φ{a} & z(φ{z}  z = a) & ψ{a} by (rule "&I")
1693    AOT_thus x(φ{x} & z(φ{z}  z = x) & ψ{x}) by (rule "∃I")
1694  next
1695    AOT_assume x(φ{x} & z(φ{z}  z = x) & ψ{x})
1696    then AOT_obtain b where g: φ{b} & z(φ{z}  z = b) & ψ{b} using "instantiation"[rotated] by blast
1697    AOT_hence h: b = ιx φ{x}  (φ{b} & z(φ{z}  z = b)) using b "∀E" by blast
1698    AOT_have φ{b} & z(φ{z}  z = b) and j: ψ{b} using g "&E" by blast+
1699    AOT_hence b = ιx φ{x} using h "≡E" by blast
1700    AOT_thus ψ{ιx φ{x}} using j "rule=E" by blast
1701  qed
1702qed
1703end
1704
1705(* TODO: this nicely shows off using locales with the embedding, but maybe there is still a nicer way *)
1706(* TODO: sledgehammer tends to refer to ψ_denotes_asm in these instantiation instead of referring
1707         to cqt:5:a - should be fixed *)
1708interpretation "russell-axiom[exe,1]": russel_axiom λ κ . «[Π]κ»
1709  by standard (metis "cqt:5:a[1]"[axiom_inst, THEN "→E"] "&E"(2))
1710interpretation "russell-axiom[exe,2,1,1]": russel_axiom λ κ . «[Π]κκ'»
1711  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E")
1712interpretation "russell-axiom[exe,2,1,2]": russel_axiom λ κ . «[Π]κ'κ»
1713  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1714interpretation "russell-axiom[exe,2,2]": russel_axiom λ κ . «[Π]κκ»
1715  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1716interpretation "russell-axiom[exe,3,1,1]": russel_axiom λ κ . «[Π]κκ'κ''»
1717  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1718interpretation "russell-axiom[exe,3,1,2]": russel_axiom λ κ . «[Π]κ'κκ''»
1719  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1720interpretation "russell-axiom[exe,3,1,3]": russel_axiom λ κ . «[Π]κ'κ''κ»
1721  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1722interpretation "russell-axiom[exe,3,2,1]": russel_axiom λ κ . «[Π]κκκ'»
1723  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1724interpretation "russell-axiom[exe,3,2,2]": russel_axiom λ κ . «[Π]κκ'κ»
1725  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1726interpretation "russell-axiom[exe,3,2,3]": russel_axiom λ κ . «[Π]κ'κκ»
1727  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1728interpretation "russell-axiom[exe,3,3]": russel_axiom λ κ . «[Π]κκκ»
1729  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1730
1731interpretation "russell-axiom[enc,1]": russel_axiom λ κ . «κ[Π]»
1732  by standard (metis "cqt:5:b[1]"[axiom_inst, THEN "→E"] "&E"(2))
1733interpretation "russell-axiom[enc,2,1]": russel_axiom λ κ . «κκ'[Π]»
1734  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E")
1735interpretation "russell-axiom[enc,2,2]": russel_axiom λ κ . «κ'κ[Π]»
1736  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1737interpretation "russell-axiom[enc,2,3]": russel_axiom λ κ . «κκ[Π]»
1738  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1739interpretation "russell-axiom[enc,3,1,1]": russel_axiom λ κ . «κκ'κ''[Π]»
1740  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1741interpretation "russell-axiom[enc,3,1,2]": russel_axiom λ κ . «κ'κκ''[Π]»
1742  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1743interpretation "russell-axiom[enc,3,1,3]": russel_axiom λ κ . «κ'κ''κ[Π]»
1744  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1745interpretation "russell-axiom[enc,3,2,1]": russel_axiom λ κ . «κκκ'[Π]»
1746  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1747interpretation "russell-axiom[enc,3,2,2]": russel_axiom λ κ . «κκ'κ[Π]»
1748  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1749interpretation "russell-axiom[enc,3,2,3]": russel_axiom λ κ . «κ'κκ[Π]»
1750  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1751interpretation "russell-axiom[enc,3,3]": russel_axiom λ κ . «κκκ[Π]»
1752  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1753
1754AOT_act_theorem "1-exists:1": ιx φ{x}  ∃!x φ{x}
1755proof(rule "≡I"; rule "→I")
1756  AOT_assume ιx φ{x}
1757  AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1758  then AOT_obtain a where a = ιx φ{x} using "instantiation"[rotated] by blast
1759  AOT_hence φ{a} & z (φ{z}  z = a) using hintikka "≡E" by blast
1760  AOT_hence x (φ{x} & z (φ{z}  z = x)) by (rule "∃I")
1761  AOT_thus ∃!x φ{x} using "uniqueness:1"[THEN "≡dfI"] by blast
1762next
1763  AOT_assume ∃!x φ{x}
1764  AOT_hence x (φ{x} & z (φ{z}  z = x))
1765    using "uniqueness:1"[THEN "≡dfE"] by blast
1766  then AOT_obtain b where φ{b} & z (φ{z}  z = b) using "instantiation"[rotated] by blast
1767  AOT_hence b = ιx φ{x} using hintikka "≡E" by blast
1768  AOT_thus ιx φ{x} by (metis "t=t-proper:2" "vdash-properties:6")
1769qed
1770
1771AOT_act_theorem "1-exists:2": y(y=ιx φ{x})  ∃!x φ{x}
1772  using "1-exists:1" "free-thms:1" "≡E"(6) by blast
1773
1774AOT_act_theorem "y-in:1": x = ιx φ{x}  φ{x}
1775  using "&E"(1) "→I" hintikka "≡E"(1) by blast
1776
1777AOT_act_theorem "y-in:2": z = ιx φ{x}  φ{z} using "y-in:1". (* TODO: same as above *)
1778
1779AOT_act_theorem "y-in:3": ιx φ{x}  φ{ιx φ{x}}
1780proof(rule "→I")
1781  AOT_assume ιx φ{x}
1782  AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1783  then AOT_obtain a where a = ιx φ{x} using "instantiation"[rotated] by blast
1784  moreover AOT_have φ{a} using calculation hintikka "≡E"(1) "&E" by blast
1785  ultimately AOT_show φ{ιx φ{x}} using "rule=E" by blast
1786qed
1787
1788AOT_act_theorem "y-in:4": y (y = ιx φ{x})  φ{ιx φ{x}}
1789  using "y-in:3"[THEN "→E"] "free-thms:1"[THEN "≡E"(2)] "→I" by blast
1790
1791
1792AOT_theorem "act-quant-nec": β (𝒜φ{β}  β = α)  β(𝒜𝒜φ{β}  β = α)
1793proof(rule "≡I"; rule "→I")
1794  AOT_assume β (𝒜φ{β}  β = α)
1795  AOT_hence 𝒜φ{β}  β = α for β using "∀E" by blast
1796  AOT_hence 𝒜𝒜φ{β}  β = α for β 
1797    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1798  AOT_thus β(𝒜𝒜φ{β}  β = α)
1799    by (rule "∀I")
1800next
1801  AOT_assume β(𝒜𝒜φ{β}  β = α)
1802  AOT_hence 𝒜𝒜φ{β}  β = α for β using "∀E" by blast
1803  AOT_hence 𝒜φ{β}  β = α for β
1804    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1805  AOT_thus β (𝒜φ{β}  β = α)
1806    by (rule "∀I")
1807qed
1808
1809AOT_theorem "equi-desc-descA:1": x = ιx φ{x}  x = ιx(𝒜φ{x})
1810proof -
1811  AOT_have x = ιx φ{x}  z (𝒜φ{z}  z = x)  using descriptions[axiom_inst] by blast
1812  also AOT_have ...  z (𝒜𝒜φ{z}  z = x)
1813  proof(rule "≡I"; rule "→I"; rule "∀I")
1814    AOT_assume z (𝒜φ{z}  z = x)
1815    AOT_hence 𝒜φ{a}  a = x for a using "∀E" by blast
1816    AOT_thus 𝒜𝒜φ{a}  a = x for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1817  next
1818    AOT_assume z (𝒜𝒜φ{z}  z = x)
1819    AOT_hence 𝒜𝒜φ{a}  a = x for a using "∀E" by blast
1820    AOT_thus 𝒜φ{a}  a = x for a  by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1821  qed
1822  also AOT_have ...  x = ιx(𝒜φ{x})
1823    using "Commutativity of ≡"[THEN "≡E"(1)] descriptions[axiom_inst] by fast
1824  finally show ?thesis .
1825qed
1826
1827AOT_theorem "equi-desc-descA:2": ιx φ{x}  ιx φ{x} = ιx(𝒜φ{x})
1828proof(rule "→I")
1829  AOT_assume ιx φ{x}
1830  AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1831  then AOT_obtain a where a = ιx φ{x} using "instantiation"[rotated] by blast
1832  moreover AOT_have a = ιx(𝒜φ{x}) using calculation "equi-desc-descA:1"[THEN "≡E"(1)] by blast
1833  ultimately AOT_show ιx φ{x} = ιx(𝒜φ{x}) using "rule=E" by fast
1834qed
1835
1836AOT_theorem "nec-hintikka-scheme": x = ιx φ{x}  𝒜φ{x} & z(𝒜φ{z}  z = x)
1837proof -
1838  AOT_have x = ιx φ{x}  z(𝒜φ{z}  z = x) using descriptions[axiom_inst] by blast
1839  also AOT_have   (𝒜φ{x} & z(𝒜φ{z}  z = x))
1840    using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" by fast
1841  finally show ?thesis.
1842qed
1843
1844AOT_theorem "equiv-desc-eq:1": 𝒜x(φ{x}  ψ{x})  x (x = ιx φ{x}  x = ιx ψ{x})
1845proof(rule "→I"; rule "∀I")
1846  fix β
1847  AOT_assume 𝒜x(φ{x}  ψ{x})
1848  AOT_hence 𝒜(φ{x}  ψ{x}) for x using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] "∀E"(2) by blast
1849  AOT_hence 0: 𝒜φ{x}  𝒜ψ{x} for x by (metis "Act-Basic:5" "≡E"(1))
1850  AOT_have β = ιx φ{x}  𝒜φ{β} & z(𝒜φ{z}  z = β) using "nec-hintikka-scheme" by blast
1851  also AOT_have ...  𝒜ψ{β} & z(𝒜ψ{z}  z = β)
1852  proof (rule "≡I"; rule "→I")
1853    AOT_assume 1: 𝒜φ{β} & z(𝒜φ{z}  z = β)
1854    AOT_hence 𝒜φ{z}  z = β for z using "&E" "∀E" by blast
1855    AOT_hence 𝒜ψ{z}  z = β for z using 0 "≡E" "→I" "→E" by metis
1856    AOT_hence z(𝒜ψ{z}  z = β) using "∀I" by fast
1857    moreover AOT_have 𝒜ψ{β} using "&E" 0[THEN "≡E"(1)] 1 by blast
1858    ultimately AOT_show 𝒜ψ{β} & z(𝒜ψ{z}  z = β) using "&I" by blast
1859  next
1860    AOT_assume 1: 𝒜ψ{β} & z(𝒜ψ{z}  z = β)
1861    AOT_hence 𝒜ψ{z}  z = β for z using "&E" "∀E" by blast
1862    AOT_hence 𝒜φ{z}  z = β for z using 0 "≡E" "→I" "→E" by metis
1863    AOT_hence z(𝒜φ{z}  z = β) using "∀I" by fast
1864    moreover AOT_have 𝒜φ{β} using "&E" 0[THEN "≡E"(2)] 1 by blast
1865    ultimately AOT_show 𝒜φ{β} & z(𝒜φ{z}  z = β) using "&I" by blast
1866  qed
1867  also AOT_have ...  β = ιx ψ{x}
1868    using "Commutativity of ≡"[THEN "≡E"(1)] "nec-hintikka-scheme" by blast
1869  finally AOT_show β = ιx φ{x}  β = ιx ψ{x} .
1870qed
1871
1872AOT_theorem "equiv-desc-eq:2": ιx φ{x} & 𝒜x(φ{x}  ψ{x})  ιx φ{x} = ιx ψ{x}
1873proof(rule "→I")
1874  AOT_assume ιx φ{x} & 𝒜x(φ{x}  ψ{x})
1875  AOT_hence 0: y (y = ιx φ{x}) and
1876            1: x (x = ιx φ{x}  x = ιx ψ{x})
1877    using "&E" "free-thms:1"[THEN "≡E"(1)] "equiv-desc-eq:1" "→E" by blast+
1878  then AOT_obtain a where a = ιx φ{x} using "instantiation"[rotated] by blast
1879  moreover AOT_have a = ιx ψ{x} using calculation 1 "∀E" "≡E"(1) by fast
1880  ultimately AOT_show ιx φ{x} = ιx ψ{x}
1881    using "rule=E" by fast
1882qed
1883
1884AOT_theorem "equiv-desc-eq:3": ιx φ{x} & x(φ{x}  ψ{x})  ιx φ{x} = ιx ψ{x}
1885  using "→I" "equiv-desc-eq:2"[THEN "→E", OF "&I"] "&E" "nec-imp-act"[THEN "→E"] by metis
1886
1887(* Note: this is a special case of "exist-nec" *)
1888AOT_theorem "equiv-desc-eq:4": ιx φ{x}  ιx φ{x}
1889proof(rule "→I")
1890  AOT_assume ιx φ{x}
1891  AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1892  then AOT_obtain a where a = ιx φ{x} using "instantiation"[rotated] by blast
1893  AOT_thus ιx φ{x}
1894    using "ex:2:a" "rule=E" by fast
1895qed
1896
1897AOT_theorem "equiv-desc-eq:5": ιx φ{x}  y (y = ιx φ{x})
1898proof(rule "→I")
1899  AOT_assume ιx φ{x}
1900  AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1901  then AOT_obtain a where a = ιx φ{x} using "instantiation"[rotated] by blast
1902  AOT_hence (a = ιx φ{x}) by (metis "id-nec:2" "vdash-properties:10")
1903  AOT_thus y (y = ιx φ{x}) by (rule "∃I")
1904qed
1905
1906AOT_act_theorem "equiv-desc-eq2:1": x (φ{x}  ψ{x})  x (x = ιx φ{x}  x = ιx ψ{x})
1907  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:1"[THEN "→E"]
1908        "RA[1]" "deduction-theorem" by blast
1909
1910AOT_act_theorem "equiv-desc-eq2:2": ιx φ{x} & x (φ{x}  ψ{x})  ιx φ{x} = ιx ψ{x}
1911  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:2"[THEN "→E", OF "&I"]
1912        "RA[1]" "deduction-theorem" "&E" by metis
1913
1914context russel_axiom
1915begin
1916AOT_theorem "nec-russell-axiom": ψ{ιx φ{x}}  x(𝒜φ{x} & z(𝒜φ{z}  z = x) & ψ{x})
1917proof -
1918  AOT_have b: x (x = ιx φ{x}  (𝒜φ{x} & z(𝒜φ{z}  z = x)))
1919    using "nec-hintikka-scheme" "∀I" by fast
1920  show ?thesis
1921  proof(rule "≡I"; rule "→I")
1922    AOT_assume c: ψ{ιx φ{x}}
1923    AOT_hence d: ιx φ{x} using ψ_denotes_asm by blast
1924    AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1925    then AOT_obtain a where a_def: a = ιx φ{x} using "instantiation"[rotated] by blast
1926    moreover AOT_have a = ιx φ{x}  (𝒜φ{a} & z(𝒜φ{z}  z = a)) using b "∀E" by blast
1927    ultimately AOT_have 𝒜φ{a} & z(𝒜φ{z}  z = a) using "≡E" by blast
1928    moreover AOT_have ψ{a}
1929    proof - 
1930      AOT_have 1: xy(x = y  y = x)
1931        by (simp add: "id-eq:2" "universal-cor")
1932      AOT_have a = ιx φ{x}   ιx φ{x} = a
1933        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1934           (auto simp: d "universal-cor" 1)
1935      AOT_thus ψ{a}
1936        using a_def c "rule=E" "→E" by metis
1937    qed
1938    ultimately AOT_have 𝒜φ{a} & z(𝒜φ{z}  z = a) & ψ{a} by (rule "&I")
1939    AOT_thus x(𝒜φ{x} & z(𝒜φ{z}  z = x) & ψ{x}) by (rule "∃I")
1940  next
1941    AOT_assume x(𝒜φ{x} & z(𝒜φ{z}  z = x) & ψ{x})
1942    then AOT_obtain b where g: 𝒜φ{b} & z(𝒜φ{z}  z = b) & ψ{b} using "instantiation"[rotated] by blast
1943    AOT_hence h: b = ιx φ{x}  (𝒜φ{b} & z(𝒜φ{z}  z = b)) using b "∀E" by blast
1944    AOT_have 𝒜φ{b} & z(𝒜φ{z}  z = b) and j: ψ{b} using g "&E" by blast+
1945    AOT_hence b = ιx φ{x} using h "≡E" by blast
1946    AOT_thus ψ{ιx φ{x}} using j "rule=E" by blast
1947  qed
1948qed
1949end
1950
1951AOT_theorem "actual-desc:1": ιx φ{x}  ∃!x 𝒜φ{x}
1952proof (rule "≡I"; rule "→I")
1953  AOT_assume ιx φ{x}
1954  AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1955  then AOT_obtain a where a = ιx φ{x} using "instantiation"[rotated] by blast
1956  moreover AOT_have a = ιx φ{x}  z(𝒜φ{z}  z = a)
1957    using descriptions[axiom_inst] by blast
1958  ultimately AOT_have z(𝒜φ{z}  z = a)
1959    using "≡E" by blast
1960  AOT_hence xz(𝒜φ{z}  z = x) by (rule "∃I")
1961  AOT_thus ∃!x 𝒜φ{x}
1962    using "uniqueness:2"[THEN "≡E"(2)] by fast
1963next
1964  AOT_assume ∃!x 𝒜φ{x}
1965  AOT_hence xz(𝒜φ{z}  z = x)
1966    using "uniqueness:2"[THEN "≡E"(1)] by fast
1967  then AOT_obtain a where z(𝒜φ{z}  z = a) using "instantiation"[rotated] by blast
1968  moreover AOT_have a = ιx φ{x}  z(𝒜φ{z}  z = a)
1969    using descriptions[axiom_inst] by blast
1970  ultimately AOT_have a = ιx φ{x} using "≡E" by blast
1971  AOT_thus ιx φ{x} by (metis "t=t-proper:2" "vdash-properties:6")
1972qed
1973
1974AOT_theorem "actual-desc:2": x = ιx φ{x}  𝒜φ{x}
1975  using "&E"(1) "contraposition:1[2]" "≡E"(1) "nec-hintikka-scheme" "reductio-aa:2" "vdash-properties:9" by blast
1976
1977AOT_theorem "actual-desc:3": z = ιx φ{x}  𝒜φ{z}
1978  using "actual-desc:2". (* TODO: same as above *)
1979
1980AOT_theorem "actual-desc:4": ιx φ{x}  𝒜φ{ιx φ{x}}
1981proof(rule "→I")
1982  AOT_assume ιx φ{x}
1983  AOT_hence y (y = ιx φ{x}) by (metis "rule=I:1" "existential:1")
1984  then AOT_obtain a where a = ιx φ{x} using "instantiation"[rotated] by blast
1985  AOT_thus 𝒜φ{ιx φ{x}}
1986    using "actual-desc:2" "rule=E" "→E" by fast
1987qed
1988
1989(* TODO: take another look at proof in PLM *)
1990AOT_theorem "actual-desc:5": ιx φ{x} = ιx ψ{x}  𝒜x(φ{x}  ψ{x})
1991proof(rule "→I")
1992  AOT_assume 0: ιx φ{x} = ιx ψ{x}
1993  AOT_hence φ_down: ιx φ{x} and ψ_down: ιx ψ{x}
1994    using "t=t-proper:1" "t=t-proper:2" "vdash-properties:6" by blast+
1995  AOT_hence y (y = ιx φ{x}) and y (y = ιx ψ{x}) by (metis "rule=I:1" "existential:1")+
1996  then AOT_obtain a and b where a_eq: a = ιx φ{x} and b_eq: b = ιx ψ{x}
1997    using "instantiation"[rotated] by metis
1998
1999  AOT_have αβ (α = β  β = α) by (rule "∀I"; rule "∀I"; rule "id-eq:2")
2000  AOT_hence β (ιx φ{x} = β  β = ιx φ{x})
2001    using "∀E" φ_down by blast
2002  AOT_hence ιx φ{x} = ιx ψ{x}  ιx ψ{x} = ιx φ{x}
2003    using "∀E" ψ_down by blast
2004  AOT_hence 1: ιx ψ{x} = ιx φ{x} using 0
2005    "→E" by blast
2006
2007  AOT_have 𝒜φ{x}  𝒜ψ{x} for x
2008  proof(rule "≡I"; rule "→I")
2009    AOT_assume 𝒜φ{x}
2010    moreover AOT_have 𝒜φ{x}  x = a for x
2011      using "nec-hintikka-scheme"[THEN "≡E"(1), OF a_eq, THEN "&E"(2)] "∀E" by blast
2012    ultimately AOT_have x = a using "→E" by blast
2013    AOT_hence x = ιx φ{x} using a_eq "rule=E" by blast
2014    AOT_hence x = ιx ψ{x} using 0 "rule=E" by blast
2015    AOT_thus 𝒜ψ{x} by (metis "actual-desc:3" "vdash-properties:6")
2016  next
2017    AOT_assume 𝒜ψ{x}
2018    moreover AOT_have 𝒜ψ{x}  x = b for x
2019      using "nec-hintikka-scheme"[THEN "≡E"(1), OF b_eq, THEN "&E"(2)] "∀E" by blast
2020    ultimately AOT_have x = b using "→E" by blast
2021    AOT_hence x = ιx ψ{x} using b_eq "rule=E" by blast
2022    AOT_hence x = ιx φ{x} using 1 "rule=E" by blast
2023    AOT_thus 𝒜φ{x} by (metis "actual-desc:3" "vdash-properties:6")
2024  qed
2025  AOT_hence 𝒜(φ{x}  ψ{x}) for x by (metis "Act-Basic:5" "≡E"(2))
2026  AOT_hence x 𝒜(φ{x}  ψ{x}) by (rule "∀I")
2027  AOT_thus 𝒜x (φ{x}  ψ{x})
2028    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
2029qed    
2030
2031AOT_theorem "!box-desc:1": ∃!x φ{x}  y (y = ιx φ{x}  φ{y})
2032proof(rule "→I")
2033  AOT_assume ∃!x φ{x}
2034  AOT_hence ζ: x (φ{x} & z (φ{z}  z = x))
2035    using "uniqueness:1"[THEN "≡dfE"] by blast
2036  then AOT_obtain b where θ: φ{b} & z (φ{z}  z = b) using "instantiation"[rotated] by blast
2037  AOT_show y (y = ιx φ{x}  φ{y})
2038  proof(rule GEN; rule "→I")
2039    fix y
2040    AOT_assume y = ιx φ{x}
2041    AOT_hence 𝒜φ{y} & z (𝒜φ{z}  z = y) using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
2042    AOT_hence 𝒜φ{b}  b = y using "&E" "∀E" by blast
2043    moreover AOT_have 𝒜φ{b} using θ[THEN "&E"(1)]  by (metis "nec-imp-act" "→E")
2044    ultimately AOT_have b = y using "→E" by blast
2045    moreover AOT_have φ{b} using θ[THEN "&E"(1)]  by (metis "qml:2"[axiom_inst] "→E") 
2046    ultimately AOT_show φ{y} using "rule=E" by blast
2047  qed
2048qed
2049
2050AOT_theorem "!box-desc:2": x (φ{x}  φ{x})  (∃!x φ{x}  y (y = ιx φ{x}  φ{y}))
2051proof(rule "→I"; rule "→I")
2052  AOT_assume x (φ{x}  φ{x})
2053  moreover AOT_assume ∃!x φ{x}
2054  ultimately AOT_have ∃!x φ{x}
2055    using "nec-exist-!"[THEN "→E", THEN "→E"] by blast
2056  AOT_thus y (y = ιx φ{x}  φ{y})
2057    using "!box-desc:1" "→E" by blast
2058qed
2059
2060AOT_theorem "dr-alphabetic-thm": ιν φ{ν}  ιν φ{ν} = ιμ φ{μ} (* TODO: vacuous *)
2061  by (simp add: "rule=I:1" "→I")
2062
2063AOT_theorem "RM:1[prem]": assumes Γ  φ  ψ shows Γ  φ  ψ
2064proof -
2065  AOT_have Γ  (φ  ψ) using "RN[prem]" assms by blast
2066  AOT_thus Γ  φ  ψ by (metis "qml:1"[axiom_inst] "→E")
2067qed
2068
2069AOT_theorem "RM:1": assumes  φ  ψ shows  φ  ψ
2070  using "RM:1[prem]" assms by blast
2071
2072lemmas RM = "RM:1"
2073
2074AOT_theorem "RM:2[prem]": assumes Γ  φ  ψ shows Γ  φ  ψ
2075proof -
2076  AOT_have Γ  ¬ψ  ¬φ using assms 
2077    by (simp add: "contraposition:1[1]")
2078  AOT_hence Γ  ¬ψ  ¬φ using "RM:1[prem]" by blast
2079  AOT_thus Γ  φ  ψ
2080    by (meson "≡dfE" "≡dfI" "conventions:5" "deduction-theorem" "modus-tollens:1")
2081qed
2082
2083AOT_theorem "RM:2": assumes  φ  ψ shows  φ  ψ
2084  using "RM:2[prem]" assms by blast
2085
2086lemmas "RM◇" = "RM:2"
2087
2088AOT_theorem "RM:3[prem]": assumes Γ  φ  ψ shows Γ  φ  ψ
2089proof -
2090  AOT_have Γ  φ  ψ and Γ  ψ  φ using assms "≡E" "→I" by metis+
2091  AOT_hence Γ  φ  ψ and Γ  ψ  φ using "RM:1[prem]" by metis+
2092  AOT_thus Γ  φ  ψ
2093    by (simp add: "≡I")
2094qed
2095
2096AOT_theorem "RM:3": assumes  φ  ψ shows  φ  ψ
2097  using "RM:3[prem]" assms by blast
2098
2099lemmas RE = "RM:3"
2100
2101AOT_theorem "RM:4[prem]": assumes Γ  φ  ψ shows Γ  φ  ψ
2102proof -
2103  AOT_have Γ  φ  ψ and Γ  ψ  φ using assms "≡E" "→I" by metis+
2104  AOT_hence Γ  φ  ψ and Γ  ψ  φ using "RM:2[prem]" by metis+
2105  AOT_thus Γ  φ  ψ by (simp add: "≡I")
2106qed
2107
2108AOT_theorem "RM:4": assumes  φ  ψ shows  φ  ψ
2109  using "RM:4[prem]" assms by blast
2110
2111lemmas "RE◇" = "RM:4"
2112
2113AOT_theorem "KBasic:1": φ  (ψ  φ)
2114  by (simp add: RM "pl:1"[axiom_inst])
2115
2116AOT_theorem "KBasic:2": ¬φ  (φ  ψ)
2117  by (simp add: RM "useful-tautologies:3")
2118
2119AOT_theorem "KBasic:3": (φ & ψ)  (φ & ψ)
2120proof (rule "≡I"; rule "→I")
2121  AOT_assume (φ & ψ)
2122  AOT_thus φ & ψ
2123    by (meson RM "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2124next
2125  AOT_have φ  (ψ  (φ & ψ)) by (simp add: "RM:1" Adjunction)
2126  AOT_hence φ  (ψ  (φ & ψ))  by (metis "Hypothetical Syllogism" "qml:1"[axiom_inst])
2127  moreover AOT_assume φ & ψ
2128  ultimately AOT_show (φ & ψ)
2129    using "→E" "&E" by blast
2130qed
2131
2132AOT_theorem "KBasic:4": (φ  ψ)  ((φ  ψ) & (ψ  φ))
2133proof -
2134  AOT_have θ: ((φ  ψ) & (ψ  φ))  ((φ  ψ) & (ψ  φ))
2135    by (fact "KBasic:3")
2136  AOT_modally_strict {
2137    AOT_have (φ  ψ)  ((φ  ψ) & (ψ  φ))
2138      by (fact "conventions:3"[THEN "≡Df"])
2139  }
2140  AOT_hence ξ: (φ  ψ)  ((φ  ψ) & (ψ  φ))
2141    by (rule RE)
2142  with ξ and θ AOT_show (φ  ψ)  ((φ  ψ) & (ψ  φ))
2143    using "≡E"(5) by blast
2144qed
2145
2146AOT_theorem "KBasic:5": ((φ  ψ) & (ψ  φ))  (φ  ψ)
2147proof -
2148  AOT_have (φ  ψ)  (φ  ψ)
2149    by (fact "qml:1"[axiom_inst])
2150  moreover AOT_have (ψ  φ)  (ψ  φ)
2151    by (fact "qml:1"[axiom_inst])
2152  ultimately AOT_have ((φ  ψ) & (ψ  φ))  ((φ  ψ) & (ψ  φ))
2153    by (metis "&I" MP "Double Composition")
2154  moreover AOT_have ((φ  ψ) & (ψ  φ))  (φ  ψ)
2155    using "conventions:3"[THEN "≡dfI"] "→I" by blast
2156  ultimately AOT_show ((φ  ψ) & (ψ  φ))  (φ  ψ)
2157    by (metis "Hypothetical Syllogism")
2158qed
2159
2160AOT_theorem "KBasic:6": (φ ψ)  (φ  ψ)
2161  using "KBasic:4" "KBasic:5" "deduction-theorem" "≡E"(1) "vdash-properties:10" by blast
2162AOT_theorem "KBasic:7": ((φ & ψ)  (¬φ & ¬ψ))  (φ  ψ)
2163proof (rule "→I"; drule "∨E"(1); (rule "→I")?)
2164  AOT_assume φ & ψ
2165  AOT_hence φ and ψ using "&E" by blast+
2166  AOT_hence (φ  ψ) and (ψ  φ) using "KBasic:1" "→E" by blast+
2167  AOT_hence (φ  ψ) & (ψ  φ) using "&I" by blast
2168  AOT_thus (φ  ψ)  by (metis "KBasic:4" "≡E"(2))
2169next
2170  AOT_assume ¬φ & ¬ψ
2171  AOT_hence 0: (¬φ & ¬ψ) using "KBasic:3"[THEN "≡E"(2)] by blast
2172  AOT_modally_strict {
2173    AOT_have (¬φ & ¬ψ)  (φ  ψ)
2174      by (metis "&E"(1) "&E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
2175  }
2176  AOT_hence (¬φ & ¬ψ)  (φ  ψ)
2177    by (rule RM)
2178  AOT_thus (φ  ψ) using 0 "→E" by blast
2179qed(auto)
2180
2181AOT_theorem "KBasic:8": (φ & ψ)  (φ  ψ)
2182  by (meson "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I")
2183AOT_theorem "KBasic:9": (¬φ & ¬ψ)  (φ  ψ)
2184  by (metis "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I" "raa-cor:4")
2185AOT_theorem "KBasic:10": φ  ¬¬φ
2186  by (simp add: "RM:3" "oth-class-taut:3:b")
2187AOT_theorem "KBasic:11": ¬φ  ¬φ
2188proof (rule "≡I"; rule "→I")
2189  AOT_show ¬φ if ¬φ
2190    using that "≡dfI" "conventions:5" "KBasic:10" "≡E"(3) by blast
2191next
2192  AOT_show ¬φ if ¬φ
2193    using "≡dfE" "conventions:5" "KBasic:10" "≡E"(4) that by blast
2194qed
2195AOT_theorem "KBasic:12": φ  ¬¬φ
2196proof (rule "≡I"; rule "→I")
2197  AOT_show ¬¬φ if φ
2198    using "¬¬I" "KBasic:11" "≡E"(3) that by blast
2199next
2200  AOT_show φ if ¬¬φ
2201  using "KBasic:11" "≡E"(1) "reductio-aa:1" that by blast
2202qed
2203AOT_theorem "KBasic:13": (φ  ψ)  (φ  ψ)
2204proof -
2205  AOT_have φ  ψ  φ  ψ by blast
2206  AOT_hence (φ  ψ)  φ  ψ
2207    using "RM:2[prem]" by blast
2208  AOT_thus (φ  ψ)  (φ  ψ) using "→I" by blast
2209qed
2210lemmas "K◇" = "KBasic:13"
2211AOT_theorem "KBasic:14": φ  ¬¬φ
2212  by (meson "RE◇" "KBasic:11" "KBasic:12" "≡E"(6) "oth-class-taut:3:a")
2213AOT_theorem "KBasic:15": (φ  ψ)  (φ  ψ)
2214proof -
2215  AOT_modally_strict {
2216    AOT_have φ  (φ  ψ) and ψ  (φ  ψ)
2217      by (auto simp: "Disjunction Addition"(1) "Disjunction Addition"(2))
2218  }
2219  AOT_hence φ  (φ  ψ) and ψ  (φ  ψ)
2220    using RM by blast+
2221  AOT_thus (φ  ψ)  (φ  ψ)
2222    by (metis "∨E"(1) "deduction-theorem")
2223qed
2224
2225AOT_theorem "KBasic:16": (φ & ψ)  (φ & ψ)
2226  by (meson "KBasic:13" "RM:1" Adjunction "Hypothetical Syllogism" Importation "vdash-properties:6")
2227
2228AOT_theorem "rule-sub-lem:1:a":
2229  assumes  (ψ  χ)
2230  shows  ¬ψ  ¬χ
2231  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2232        "≡E"(1) "oth-class-taut:4:b" by blast
2233
2234AOT_theorem "rule-sub-lem:1:b":
2235  assumes  (ψ  χ)
2236  shows  (ψ  Θ)  (χ  Θ)
2237  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2238  using "oth-class-taut:4:c" "vdash-properties:6" by blast
2239
2240AOT_theorem "rule-sub-lem:1:c":
2241  assumes  (ψ  χ)
2242  shows  (Θ  ψ)  (Θ  χ)
2243  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2244  using "oth-class-taut:4:d" "vdash-properties:6" by blast
2245
2246AOT_theorem "rule-sub-lem:1:d":
2247  assumes for arbitrary α:  (ψ{α}  χ{α})
2248  shows  α ψ{α}  α χ{α}
2249proof -
2250  AOT_modally_strict {
2251    AOT_have α (ψ{α}  χ{α})
2252      using "qml:2"[axiom_inst, THEN "→E", OF assms] "∀I" by fast
2253    AOT_hence 0: ψ{α}  χ{α} for α using "∀E" by blast
2254    AOT_show α ψ{α}  α χ{α}
2255    proof (rule "≡I"; rule "→I")
2256      AOT_assume α ψ{α}
2257      AOT_hence ψ{α} for α using "∀E" by blast
2258      AOT_hence χ{α} for α using 0 "≡E" by blast
2259      AOT_thus α χ{α} by (rule "∀I")
2260    next
2261      AOT_assume α χ{α}
2262      AOT_hence χ{α} for α using "∀E" by blast
2263      AOT_hence ψ{α} for α using 0 "≡E" by blast
2264      AOT_thus α ψ{α} by (rule "∀I")
2265    qed
2266  }
2267qed
2268
2269AOT_theorem "rule-sub-lem:1:e":
2270  assumes  (ψ  χ)
2271  shows   ψ]   χ]
2272  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2273  using "≡E"(1) "propositions-lemma:6" by blast
2274
2275AOT_theorem "rule-sub-lem:1:f":
2276  assumes  (ψ  χ)
2277  shows  𝒜ψ  𝒜χ
2278  using "qml:2"[axiom_inst, THEN "→E", OF assms, THEN "RA[2]"]
2279  by (metis "Act-Basic:5" "≡E"(1))
2280
2281AOT_theorem "rule-sub-lem:1:g":
2282  assumes  (ψ  χ)
2283  shows  ψ  χ
2284  using "KBasic:6" assms "vdash-properties:6" by blast
2285
2286text‹Note that instead of deriving @{text "rule-sub-lem:2"}, @{text "rule-sub-lem:3"}, @{text "rule-sub-lem:4"},
2287     and @{text "rule-sub-nec"}, we construct substitution methods instead.›
2288
2289class AOT_subst =
2290  fixes AOT_subst :: "('a  𝗈)  bool"
2291    and AOT_subst_cond :: "'a  'a  bool"
2292  assumes AOT_subst: "AOT_subst φ  AOT_subst_cond ψ χ  [v  «φ ψ»  «φ χ»]"
2293
2294named_theorems AOT_substI
2295
2296instantiation 𝗈 :: AOT_subst
2297begin
2298
2299inductive AOT_subst_𝗈 where
2300  AOT_subst_𝗈_id[AOT_substI]: "AOT_subst_𝗈 (λφ. φ)"
2301| AOT_subst_𝗈_const[AOT_substI]: "AOT_subst_𝗈 (λφ. ψ)"
2302| AOT_subst_𝗈_not[AOT_substI]: "AOT_subst_𝗈 Θ  AOT_subst_𝗈 (λ φ. «¬Θ{φ}»)"
2303| AOT_subst_𝗈_imp[AOT_substI]: "AOT_subst_𝗈 Θ  AOT_subst_𝗈 Ξ  AOT_subst_𝗈 (λ φ. «Θ{φ}  Ξ{φ}»)"
2304| AOT_subst_𝗈_lambda0[AOT_substI]: "AOT_subst_𝗈 Θ  AOT_subst_𝗈 (λ φ. (AOT_lambda0 (Θ φ)))"
2305| AOT_subst_𝗈_act[AOT_substI]: "AOT_subst_𝗈 Θ  AOT_subst_𝗈 (λ φ. «𝒜Θ{φ}»)"
2306| AOT_subst_𝗈_box[AOT_substI]: "AOT_subst_𝗈 Θ  AOT_subst_𝗈 (λ φ. «Θ{φ}»)"
2307| AOT_subst_𝗈_by_def[AOT_substI]: "( ψ . AOT_model_equiv_def (Θ ψ) (Ξ ψ))  AOT_subst_𝗈 Ξ  AOT_subst_𝗈 Θ"
2308
2309definition AOT_subst_cond_𝗈 where "AOT_subst_cond_𝗈  λ ψ χ .  v . [v  ψ  χ]"
2310
2311instance
2312proof
2313  fix ψ χ :: 𝗈 and φ :: ‹𝗈  𝗈›
2314  assume cond: ‹AOT_subst_cond ψ χ
2315  assume ‹AOT_subst φ
2316  moreover AOT_have  ψ  χ using cond unfolding AOT_subst_cond_𝗈_def by blast
2317  ultimately AOT_show  φ{ψ}  φ{χ}
2318  proof (induct arbitrary: ψ χ)
2319    case AOT_subst_𝗈_id
2320    thus ?case using "≡E"(2) "oth-class-taut:4:b" "rule-sub-lem:1:a" by blast
2321  next
2322    case (AOT_subst_𝗈_const ψ)
2323    thus ?case by (simp add: "oth-class-taut:3:a")
2324  next
2325    case (AOT_subst_𝗈_not Θ)
2326    thus ?case by (simp add: RN "rule-sub-lem:1:a")
2327  next
2328    case (AOT_subst_𝗈_imp Θ Ξ)
2329    thus ?case by (meson RN "≡E"(5) "rule-sub-lem:1:b" "rule-sub-lem:1:c")
2330  next
2331    case (AOT_subst_𝗈_lambda0 Θ)
2332    thus ?case by (simp add: RN "rule-sub-lem:1:e")
2333  next
2334    case (AOT_subst_𝗈_act Θ)
2335    thus ?case by (simp add: RN "rule-sub-lem:1:f")
2336  next
2337    case (AOT_subst_𝗈_box Θ)
2338    thus ?case by (simp add: RN "rule-sub-lem:1:g")
2339  next
2340    case (AOT_subst_𝗈_by_def Θ Ξ)
2341    AOT_modally_strict {
2342      AOT_have Ξ{ψ}  Ξ{χ} using AOT_subst_𝗈_by_def by simp
2343      AOT_thus Θ{ψ}  Θ{χ}
2344        using "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ ψ] "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ χ]
2345        by (metis "≡E"(6) "oth-class-taut:3:a")
2346    }
2347  qed
2348qed
2349end
2350
2351instantiation "fun" :: (AOT_Term_id_2, AOT_subst) AOT_subst
2352begin
2353
2354definition AOT_subst_cond_fun :: "('a  'b)  ('a  'b)  bool" where
2355  "AOT_subst_cond_fun  λ φ ψ .  α . AOT_subst_cond (φ (AOT_term_of_var α)) (ψ (AOT_term_of_var α))"
2356
2357inductive AOT_subst_fun :: "(('a  'b)  𝗈)  bool" where
2358  AOT_subst_fun_const[AOT_substI]: "AOT_subst_fun (λφ. ψ)"
2359| AOT_subst_fun_id[AOT_substI]: "AOT_subst Ψ  AOT_subst_fun (λφ. Ψ (φ (AOT_term_of_var x)))"
2360| AOT_subst_fun_all[AOT_substI]: "AOT_subst Ψ  ( α . AOT_subst_fun (Θ (AOT_term_of_var α)))  AOT_subst_fun (λφ :: 'a  'b. Ψ «α «Θ (α::'a) φ»»)"
2361| AOT_subst_fun_not[AOT_substI]: "AOT_subst Ψ  AOT_subst_fun (λφ. «¬«Ψ φ»»)"
2362| AOT_subst_fun_imp[AOT_substI]: "AOT_subst Ψ  AOT_subst Θ  AOT_subst_fun (λφ. ««Ψ φ»  «Θ φ»»)"
2363| AOT_subst_fun_lambda0[AOT_substI]: "AOT_subst Θ  AOT_subst_fun (λ φ. (AOT_lambda0 (Θ φ)))"
2364| AOT_subst_fun_act[AOT_substI]: "AOT_subst Θ  AOT_subst_fun (λ φ. «𝒜«Θ φ»»)"
2365| AOT_subst_fun_box[AOT_substI]: "AOT_subst Θ  AOT_subst_fun (λ φ. ««Θ φ»»)"
2366| AOT_subst_fun_def[AOT_substI]: "( φ . AOT_model_equiv_def (Θ φ) (Ψ φ))  AOT_subst_fun Ψ  AOT_subst_fun Θ"
2367
2368instance proof
2369  fix ψ χ :: "'a  'b" and φ :: ('a  'b)  𝗈›
2370  assume ‹AOT_subst φ
2371  moreover assume cond: ‹AOT_subst_cond ψ χ
2372  ultimately AOT_show  «φ ψ»  «φ χ»
2373  proof(induct)
2374    case (AOT_subst_fun_const ψ)
2375    then show ?case by (simp add: "oth-class-taut:3:a")
2376  next
2377  case (AOT_subst_fun_id Ψ x)
2378  then show ?case by (simp add: AOT_subst AOT_subst_cond_fun_def) 
2379  next
2380  case (AOT_subst_fun_all Ψ Θ)
2381  AOT_have  (Θ{α, «ψ»}  Θ{α, «χ»}) for α
2382    using AOT_subst_fun_all.hyps(3) AOT_subst_fun_all.prems RN by presburger
2383  thus ?case using AOT_subst[OF AOT_subst_fun_all(1)]
2384    by (simp add: RN "rule-sub-lem:1:d" AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def)
2385  next
2386  case (AOT_subst_fun_not Ψ)
2387  then show ?case by (simp add: RN "rule-sub-lem:1:a")
2388  next
2389  case (AOT_subst_fun_imp Ψ Θ)
2390  then show ?case 
2391    unfolding AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def
2392    by (meson "≡E"(5) "oth-class-taut:4:c" "oth-class-taut:4:d" "vdash-properties:6")
2393  next
2394  case (AOT_subst_fun_lambda0 Θ)
2395  then show ?case by (simp add: RN "rule-sub-lem:1:e")
2396  next
2397  case (AOT_subst_fun_act Θ)
2398  then show ?case by (simp add: RN "rule-sub-lem:1:f")
2399  next
2400  case (AOT_subst_fun_box Θ)
2401  then show ?case by (simp add: RN "rule-sub-lem:1:g")
2402  next
2403  case (AOT_subst_fun_def Θ Ψ)
2404  then show ?case
2405    by (meson "df-rules-formulas[3]" "df-rules-formulas[4]" "≡I" "≡E"(5))
2406  qed
2407qed
2408end
2409
2410method_setup AOT_defI =
2411‹Scan.lift (Scan.succeed (fn ctxt => (Method.CONTEXT_METHOD (fn thms => (Context_Tactic.CONTEXT_SUBGOAL (fn (trm,int) => 
2412Context_Tactic.CONTEXT_TACTIC (
2413let
2414fun findHeadConst (Const x) = SOME x
2415  | findHeadConst (A $ B) = findHeadConst A
2416  | findHeadConst _ = NONE
2417fun findDef (Const (const_name‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2418  | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2419  | findDef (Abs (a,b,c)) = findDef c
2420  | findDef _ = NONE
2421val const_opt = (findDef trm)
2422val defs = case const_opt of SOME const => List.filter (fn thm => let
2423    val concl = Thm.concl_of thm
2424    val thmconst = (findDef concl)
2425    in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2426    | _ => []
2427in
2428resolve_tac ctxt defs 1
2429end
2430)) 1)))))
2431‹Resolve AOT definitions›
2432
2433
2434method AOT_subst_intro_helper = ((rule AOT_substI
2435      | AOT_defI
2436      | (simp only: AOT_subst_cond_𝗈_def AOT_subst_cond_fun_def; ((rule allI)+)?)))
2437
2438method_setup AOT_subst = 2439Scan.option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
2440Scan.lift (Args.embedded_inner_syntax -- Args.embedded_inner_syntax
2441) -- Scan.option (Scan.lift (Args.$$$ "bound" -- Args.colon) |-- Scan.repeat1 (Scan.lift (Args.embedded_inner_syntax)))
2442>> (fn ((reversed,(raw_p,raw_q)),raw_bounds) => (fn ctxt =>
2443(Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _, prems = prems, asms = asms, concl = concl, schematics = _} =>
2444let
2445val thms = prems
2446val ctxt' = ctxt
2447val ctxt = Context_Position.set_visible false ctxt
2448val raw_bounds = case raw_bounds of SOME bounds => bounds | _ => [] 
2449val bounds = (map (fn x => Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt x))) raw_bounds
2450
2451val p = AOT_read_term @{nonterminal φ'} ctxt raw_p
2452val p = Syntax.check_term ctxt p
2453val p = fold (fn bound => fn p => let in Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,p)) end) bounds p
2454val p = Syntax.check_term ctxt p
2455val p_ty = Term.type_of p
2456val pat = @{const Trueprop} $ (@{const AOT_model_valid_in} $ Var (("w",0), @{typ w}) $ (Var (("φ",0), Type (type_name‹fun›, [p_ty, @{typ 𝗈}])) $ p))
2457val univ = Unify.matchers (Context.Proof ctxt) [(pat, Thm.term_of concl)]
2458val univ = hd (Seq.list_of univ) (* TODO: choose? try all? filter? *)
2459val phi = the (Envir.lookup univ (("φ",0), Type (type_name‹fun›, [p_ty, @{typ 𝗈}])))
2460
2461
2462val q = AOT_read_term @{nonterminal φ'} ctxt raw_q
2463val q = Syntax.check_term ctxt q
2464val q = fold (fn bound => fn q => let in Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,q)) end) bounds q
2465val q = Syntax.check_term ctxt q
2466
2467(* Reparse to report bounds as fixes. *)
2468val ctxt = Context_Position.restore_visible ctxt' ctxt
2469val ctxt' = ctxt
2470fun unsource str = fst (Input.source_content (Syntax.read_input str))
2471val (_,ctxt') = Proof_Context.add_fixes (map (fn str => (Binding.make (unsource str, Position.none), NONE, Mixfix.NoSyn)) raw_bounds) ctxt'
2472val _ = (map (fn x => Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt' x))) raw_bounds
2473val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_p
2474val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_q
2475
2476
2477val abs = phi
2478val abs = HOLogic.mk_Trueprop (@{const AOT_subst(_)} $ abs)
2479val abs = Syntax.check_term ctxt abs
2480
2481val substThm = Goal.prove ctxt [] [] abs (fn {context=ctxt, prems=prems} =>
2482      REPEAT (SUBGOAL (fn (trm,int) => let
2483          fun findHeadConst (Const x) = SOME x
2484            | findHeadConst (A $ B) = findHeadConst A
2485            | findHeadConst _ = NONE
2486          fun findDef (Const (const_name‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2487            | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2488            | findDef (Abs (a,b,c)) = findDef c
2489            | findDef _ = NONE
2490          val const_opt = (findDef trm)
2491          val defs = case const_opt of SOME const => List.filter (fn thm => let
2492              val concl = Thm.concl_of thm
2493              val thmconst = (findDef concl)
2494              in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2495              | _ => []
2496          val tac = case defs of [] => safe_step_tac (ctxt addSIs @{thms AOT_substI}) 1
2497                    | _ => resolve_tac ctxt defs 1
2498        in tac end) 1)
2499  )
2500val substThm = substThm RS @{thm AOT_subst}
2501val abs = Thm.cterm_of ctxt abs
2502val substThm = case reversed of NONE =>
2503  let
2504  val substThm = Drule.instantiate_normalize ([],[((("ψ", 0), p_ty), Thm.cterm_of ctxt p),
2505          ((("χ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
2506  val substThm = substThm RS @{thm "≡E"(2)}
2507  in substThm end
2508| _ =>   let
2509  val substThm = Drule.instantiate_normalize ([],[((("χ", 0), p_ty), Thm.cterm_of ctxt p),
2510          ((("ψ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
2511  val substThm = substThm RS @{thm "≡E"(1)}
2512  in substThm end
2513
2514in
2515resolve_tac ctxt [substThm] 1
2516THEN simp_tac (ctxt addsimps [@{thm AOT_subst_cond_𝗈_def}, @{thm AOT_subst_cond_fun_def}]) 1
2517THEN (REPEAT (resolve_tac ctxt [@{thm allI}] 1))
2518THEN (TRY (resolve_tac ctxt thms 1))
2519end
2520) ctxt 1))))
2521
2522
2523thm AOT_subst
2524declare [[ML_print_depth=100]]
2525
2526AOT_theorem
2527  assumes  p  q and (q  q)
2528  shows (p  p)
2529  using assms  apply (AOT_subst p q)
2530  oops
2531AOT_theorem
2532  assumes  p  q and (q  q)
2533  shows p(q & p  p)
2534  apply (AOT_subst q & p r bound: p)
2535  oops
2536AOT_theorem
2537  assumes  p  q
2538  shows x([F]x  q)
2539  apply (AOT_subst [F]x [G]x bound: x)
2540  oops
2541
2542method AOT_subst_old for ψ::"'a::AOT_subst" and χ::"'a::AOT_subst" =
2543    (match conclusion in "[v  «φ ψ»]" for φ and v 
2544      match (φ) in "λa . ?p" fail¦ "λa . a" fail2545       ¦ _ rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2546       ; (AOT_subst_intro_helper+)?››)
2547
2548method AOT_subst_old_rev for χ::"'a::AOT_subst" and ψ::"'a::AOT_subst" =
2549    (match conclusion in "[v  «φ ψ»]" for φ and v 
2550      match (φ) in "λa . ?p" fail¦ "λa . a" fail2551       ¦ _ rule AOT_subst[where φ=φ and ψ=χ and χ=ψ, THEN "≡E"(1)]
2552       ; (AOT_subst_intro_helper+)?››)
2553
2554method AOT_subst_manual for φ::"'a::AOT_subst  𝗈" =
2555    (rule AOT_subst[where φ=φ, THEN "≡E"(2)]; (AOT_subst_intro_helper+)?)
2556
2557method AOT_subst_manual_rev for φ::"'a::AOT_subst  𝗈" =
2558    (rule AOT_subst[where φ=φ, THEN "≡E"(1)]; (AOT_subst_intro_helper+)?)
2559
2560method AOT_subst_using uses subst =
2561    (match subst in "[?w  ψ  χ]" for ψ χ  2562       match conclusion in "[v  «φ ψ»]" for φ v 2563         rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2564         ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2565
2566method AOT_subst_using_rev uses subst =
2567    (match subst in "[?w  ψ  χ]" for ψ χ  2568      match conclusion in "[v  «φ χ»]" for φ v 2569        rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(1)]
2570        ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2571
2572AOT_theorem "rule-sub-remark:1[1]": assumes  A!x  ¬E!x and ¬A!x shows ¬¬E!x
2573  by (AOT_subst (reverse) ¬E!x A!x)
2574     (auto simp: assms) 
2575
2576AOT_theorem "rule-sub-remark:1[2]": assumes  A!x  ¬E!x and  ¬¬E!x shows ¬A!x
2577  by (AOT_subst A!x ¬E!x)
2578     (auto simp: assms)
2579
2580AOT_theorem "rule-sub-remark:2[1]":
2581  assumes  [R]xy  ([R]xy & ([Q]a  ¬[Q]a)) and p  [R]xy shows p  [R]xy & ([Q]a  ¬[Q]a)
2582  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2583
2584AOT_theorem "rule-sub-remark:2[2]":
2585  assumes  [R]xy  ([R]xy & ([Q]a  ¬[Q]a)) and p  [R]xy & ([Q]a  ¬[Q]a) shows p  [R]xy
2586  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2587
2588AOT_theorem "rule-sub-remark:3[1]":
2589  assumes for arbitrary x:  A!x  ¬E!x
2590      and x A!x
2591    shows x ¬E!x
2592  by (AOT_subst (reverse) ¬E!x A!x bound: x)
2593     (auto simp: assms)
2594
2595AOT_theorem "rule-sub-remark:3[2]":
2596  assumes for arbitrary x:  A!x  ¬E!x
2597      and x ¬E!x
2598    shows x A!x
2599  by (AOT_subst A!x ¬E!x bound: x)
2600     (auto simp: assms)
2601
2602AOT_theorem "rule-sub-remark:4[1]":
2603  assumes  ¬¬[P]x  [P]x and 𝒜¬¬[P]x shows 𝒜[P]x
2604  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2605
2606AOT_theorem "rule-sub-remark:4[2]":
2607  assumes  ¬¬[P]x  [P]x and 𝒜[P]x shows 𝒜¬¬[P]x
2608  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2609
2610AOT_theorem "rule-sub-remark:5[1]":
2611  assumes  (φ  ψ)  (¬ψ  ¬φ) and (φ  ψ) shows (¬ψ  ¬φ)
2612  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2613
2614AOT_theorem "rule-sub-remark:5[2]":
2615  assumes  (φ  ψ)  (¬ψ  ¬φ) and (¬ψ  ¬φ) shows (φ  ψ) 
2616  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2617
2618AOT_theorem "rule-sub-remark:6[1]":
2619  assumes  ψ  χ and (φ  ψ) shows (φ  χ) 
2620  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2621
2622AOT_theorem "rule-sub-remark:6[2]":
2623  assumes  ψ  χ and (φ  χ) shows (φ  ψ)
2624  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2625
2626AOT_theorem "rule-sub-remark:7[1]":
2627  assumes  φ  ¬¬φ and (φ  φ) shows (¬¬φ  φ) 
2628  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2629
2630AOT_theorem "rule-sub-remark:7[2]":
2631  assumes  φ  ¬¬φ and (¬¬φ  φ) shows  (φ  φ)
2632  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2633
2634AOT_theorem "KBasic2:1": ¬φ  ¬φ
2635  by (meson "conventions:5" "contraposition:2" "Hypothetical Syllogism" "df-rules-formulas[3]"
2636            "df-rules-formulas[4]" "≡I" "useful-tautologies:1")
2637
2638AOT_theorem "KBasic2:2": (φ  ψ)  (φ  ψ)
2639proof -
2640  AOT_have (φ  ψ)  ¬(¬φ & ¬ψ)
2641    by (simp add: "RE◇" "oth-class-taut:5:b")
2642  also AOT_have   ¬(¬φ & ¬ψ)
2643    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2644  also AOT_have   ¬(¬φ & ¬ψ)
2645    using "KBasic:3" "≡E"(1) "oth-class-taut:4:b" by blast
2646  also AOT_have   ¬(¬φ & ¬ψ)
2647    using "KBasic2:1"
2648    by (AOT_subst ¬φ ¬φ; AOT_subst ¬ψ ¬ψ; auto simp: "oth-class-taut:3:a")
2649  also AOT_have   ¬¬(φ  ψ)
2650    using "≡E"(6) "oth-class-taut:3:b" "oth-class-taut:5:b" by blast
2651  also AOT_have   φ  ψ
2652    by (simp add: "≡I" "useful-tautologies:1" "useful-tautologies:2")
2653  finally show ?thesis .
2654qed
2655
2656AOT_theorem "KBasic2:3": (φ & ψ)  (φ & ψ)
2657  by (metis "RM◇" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "deduction-theorem" "modus-tollens:1" "reductio-aa:1")
2658
2659AOT_theorem "KBasic2:4": (φ  ψ)  (φ  ψ)
2660proof -
2661  AOT_have (φ  ψ)  (¬φ  ψ)
2662    by (AOT_subst φ  ψ ¬φ  ψ)
2663       (auto simp: "oth-class-taut:1:c" "oth-class-taut:3:a")
2664  also AOT_have ...  ¬φ  ψ
2665    by (simp add: "KBasic2:2")
2666  also AOT_have ...  ¬φ  ψ
2667    by (AOT_subst ¬φ ¬φ)
2668       (auto simp: "KBasic:11" "oth-class-taut:3:a")
2669  also AOT_have ...  φ  ψ
2670    using "≡E"(6) "oth-class-taut:1:c" "oth-class-taut:3:a" by blast
2671  finally show ?thesis .
2672qed
2673
2674AOT_theorem "KBasic2:5": φ  ¬¬φ
2675  using "conventions:5"[THEN "≡Df"]
2676  by (AOT_subst φ ¬¬φ; AOT_subst ¬¬φ ¬¬¬¬φ; AOT_subst (reverse) ¬¬¬φ ¬φ)
2677     (auto simp: "oth-class-taut:3:b" "oth-class-taut:3:a")
2678
2679
2680AOT_theorem "KBasic2:6": (φ  ψ)  (φ  ψ)
2681proof(rule "→I"; rule "raa-cor:1")
2682  AOT_assume (φ  ψ)
2683  AOT_hence (¬φ  ψ)
2684    using "conventions:2"[THEN "≡Df"]
2685    by (AOT_subst (reverse) ¬φ  ψ φ  ψ) simp
2686  AOT_hence 1: ¬φ  ψ using "KBasic:13" "vdash-properties:10" by blast
2687  AOT_assume ¬(φ  ψ)
2688  AOT_hence ¬φ and ¬ψ using "&E" "≡E"(1) "oth-class-taut:5:d" by blast+
2689  AOT_thus ψ & ¬ψ using "&I"(1) 1[THEN "→E"] "KBasic:11" "≡E"(4) "raa-cor:3" by blast
2690qed
2691
2692AOT_theorem "KBasic2:7": ((φ  ψ) & ¬φ)  ψ
2693proof(rule "→I"; frule "&E"(1); drule "&E"(2))
2694  AOT_assume (φ  ψ)
2695  AOT_hence 1: φ  ψ
2696    using "KBasic2:6" "∨I"(2) "∨E"(1) by blast
2697  AOT_assume ¬φ
2698  AOT_hence ¬φ using "KBasic:11" "≡E"(2) by blast
2699  AOT_thus ψ using 1 "∨E"(2) by blast
2700qed
2701
2702AOT_theorem "T-S5-fund:1": φ  φ
2703  by (meson "≡dfI" "conventions:5" "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" "qml:2"[axiom_inst])
2704lemmas "T◇" = "T-S5-fund:1"
2705
2706AOT_theorem "T-S5-fund:2": φ  φ
2707proof(rule "→I")
2708  AOT_assume φ
2709  AOT_hence ¬¬φ
2710    using "KBasic:14" "≡E"(4) "raa-cor:3" by blast
2711  moreover AOT_have ¬φ  ¬φ
2712    by (fact "qml:3"[axiom_inst])
2713  ultimately AOT_have ¬¬φ
2714    using "modus-tollens:1" by blast
2715  AOT_thus φ using "KBasic:12" "≡E"(2) by blast
2716qed
2717lemmas "5◇" = "T-S5-fund:2"
2718
2719(* Also interestingly none of these have proofs in PLM. *)
2720AOT_theorem "Act-Sub:1": 𝒜φ  ¬𝒜¬φ
2721  by (AOT_subst 𝒜¬φ ¬𝒜φ)
2722     (auto simp: "logic-actual-nec:1"[axiom_inst] "oth-class-taut:3:b")
2723
2724AOT_theorem "Act-Sub:2": φ  𝒜φ
2725  using "conventions:5"[THEN "≡Df"]
2726  by (AOT_subst φ ¬¬φ)
2727     (metis "deduction-theorem" "≡I" "≡E"(1) "≡E"(2) "≡E"(3)
2728            "logic-actual-nec:1"[axiom_inst] "qml-act:2"[axiom_inst])
2729
2730AOT_theorem "Act-Sub:3": 𝒜φ  φ
2731  using "conventions:5"[THEN "≡Df"]
2732  by (AOT_subst φ ¬¬φ)
2733     (metis "Act-Sub:1" "deduction-theorem" "≡E"(4) "nec-imp-act" "reductio-aa:2" "→E")
2734
2735
2736AOT_theorem "Act-Sub:4": 𝒜φ  𝒜φ
2737proof (rule "≡I"; rule "→I")
2738  AOT_assume 𝒜φ
2739  AOT_thus 𝒜φ using "T◇" "vdash-properties:10" by blast
2740next
2741  AOT_assume 𝒜φ
2742  AOT_hence ¬¬𝒜φ
2743    using "≡dfE" "conventions:5" by blast
2744  AOT_hence ¬𝒜¬φ
2745    by (AOT_subst 𝒜¬φ ¬𝒜φ)
2746       (simp add: "logic-actual-nec:1"[axiom_inst])
2747  AOT_thus 𝒜φ
2748      using "Act-Basic:1" "Act-Basic:6" "∨E"(3) "≡E"(4) "reductio-aa:1" by blast
2749qed
2750
2751AOT_theorem "Act-Sub:5": 𝒜φ  𝒜φ
2752  by (metis "Act-Sub:2" "Act-Sub:3" "Act-Sub:4" "deduction-theorem" "≡E"(1) "≡E"(2) "vdash-properties:6")
2753
2754AOT_theorem "S5Basic:1": φ  φ
2755  by (simp add: "≡I" "qml:2" "qml:3" "vdash-properties:1[2]")
2756
2757AOT_theorem "S5Basic:2": φ  φ
2758  by (simp add: "T◇" "5◇" "≡I")
2759
2760AOT_theorem "S5Basic:3": φ  φ
2761  using "T◇" "Hypothetical Syllogism" "qml:3" "vdash-properties:1[2]" by blast
2762lemmas "B" = "S5Basic:3"
2763
2764AOT_theorem "S5Basic:4": φ  φ
2765  using "5◇" "Hypothetical Syllogism" "qml:2" "vdash-properties:1[2]" by blast
2766lemmas "B◇" = "S5Basic:4"
2767
2768AOT_theorem "S5Basic:5": φ  φ
2769  using "RM:1" "B" "5◇" "Hypothetical Syllogism" by blast
2770lemmas "4" = "S5Basic:5"
2771
2772AOT_theorem "S5Basic:6": φ  φ
2773  by (simp add: "4" "≡I" "qml:2"[axiom_inst])
2774
2775AOT_theorem "S5Basic:7": φ  φ
2776  using "conventions:5"[THEN "≡Df"] "oth-class-taut:3:b"
2777  by (AOT_subst φ ¬¬φ;
2778      AOT_subst φ ¬¬φ;
2779      AOT_subst (reverse) ¬¬¬φ ¬φ;
2780      AOT_subst (reverse) ¬φ ¬φ)
2781     (auto simp: "S5Basic:6" "if-p-then-p")
2782
2783lemmas "4◇" = "S5Basic:7"
2784
2785AOT_theorem "S5Basic:8": φ  φ
2786  by (simp add: "4◇" "T◇" "≡I")
2787
2788AOT_theorem "S5Basic:9": (φ  ψ)  (φ  ψ)
2789  apply (rule "≡I"; rule "→I")
2790  using "KBasic2:6" "5◇" "∨I"(3) "if-p-then-p" "vdash-properties:10" apply blast
2791  by (meson "KBasic:15" "4" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "con-dis-taut:7"
2792            "intro-elim:1" "Commutativity of ∨")
2793
2794AOT_theorem "S5Basic:10": (φ  ψ)  (φ  ψ)
2795(* Note: nicely this proof is entirely sledgehammer generated *)
2796proof(rule "≡I"; rule "→I")
2797  AOT_assume (φ  ψ)
2798  AOT_hence φ  ψ
2799    by (meson "KBasic2:6" "∨I"(2) "∨E"(1))
2800  AOT_thus φ  ψ
2801    by (meson "B◇" "4" "4◇" "T◇" "∨I"(3))
2802next
2803  AOT_assume φ  ψ
2804  AOT_hence φ  ψ
2805    by (meson "S5Basic:1" "B◇" "S5Basic:6" "T◇" "5◇" "∨I"(3) "intro-elim:1")
2806  AOT_thus (φ  ψ)
2807    by (meson "KBasic:15" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
2808qed
2809
2810AOT_theorem "S5Basic:11": (φ & ψ)  (φ & ψ)
2811proof -
2812  AOT_have (φ & ψ)  ¬(¬φ  ¬ψ)
2813    by (AOT_subst φ & ψ ¬(¬φ  ¬ψ))
2814       (auto simp: "oth-class-taut:5:a" "oth-class-taut:3:a")
2815  also AOT_have   ¬(¬φ  ¬ψ)
2816    by (AOT_subst ¬ψ ¬ψ)
2817       (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2818  also AOT_have   ¬(¬φ  ¬ψ)
2819    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2820  also AOT_have   ¬(¬φ  ¬ψ)
2821    using "S5Basic:9" "≡E"(1) "oth-class-taut:4:b" by blast
2822  also AOT_have   ¬(¬φ  ¬ψ)
2823    using "KBasic2:1"
2824    by (AOT_subst ¬φ ¬φ; AOT_subst ¬ψ ¬ψ)
2825       (auto simp:  "oth-class-taut:3:a")
2826  also AOT_have   φ & ψ
2827    using "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:a" by blast
2828  finally show ?thesis .
2829qed
2830
2831AOT_theorem "S5Basic:12": (φ & ψ)  (φ & ψ)
2832proof (rule "≡I"; rule "→I")
2833  AOT_assume (φ & ψ)
2834  AOT_hence φ & ψ
2835    using "KBasic2:3" "vdash-properties:6" by blast
2836  AOT_thus φ & ψ
2837    using "5◇" "&I" "&E"(1) "&E"(2) "vdash-properties:6" by blast
2838next
2839  AOT_assume φ & ψ
2840  moreover AOT_have (ψ & φ)  (φ & ψ)
2841    by (AOT_subst φ & ψ ψ & φ)
2842       (auto simp: "Commutativity of &" "KBasic:16")
2843  ultimately AOT_show (φ & ψ)
2844    by (metis "4" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2845qed
2846
2847
2848AOT_theorem "S5Basic:13": (φ  ψ)  (φ  ψ)
2849proof (rule "≡I")
2850  AOT_modally_strict {
2851    AOT_have (φ  ψ)  (φ  ψ)
2852      by (meson "KBasic:13" "B◇" "Hypothetical Syllogism" "deduction-theorem")
2853  }
2854  AOT_hence (φ  ψ)  (φ  ψ)
2855    by (rule RM)
2856  AOT_thus  (φ  ψ)  (φ  ψ)
2857    using "4" "Hypothetical Syllogism" by blast
2858next
2859  AOT_modally_strict {
2860    AOT_have (φ  ψ)  (φ  ψ)
2861      by (meson "B" "Hypothetical Syllogism" "deduction-theorem" "qml:1" "vdash-properties:1[2]")
2862  }
2863  AOT_hence  (φ  ψ)  (φ  ψ)
2864    by (rule RM)
2865  AOT_thus (φ  ψ)  (φ  ψ)
2866    using "4" "Hypothetical Syllogism" by blast
2867qed
2868
2869AOT_theorem "derived-S5-rules:1":
2870  assumes Γ  φ  ψ shows Γ  φ  ψ
2871proof -
2872  AOT_have Γ  φ  ψ
2873    using assms by (rule "RM:1[prem]")
2874  AOT_thus Γ  φ  ψ
2875    using "B" "Hypothetical Syllogism" by blast
2876qed
2877
2878AOT_theorem "derived-S5-rules:2":
2879  assumes Γ  φ  ψ shows Γ  φ  ψ
2880proof -
2881  AOT_have Γ  φ  ψ
2882    using assms by (rule "RM:2[prem]")
2883  AOT_thus Γ  φ  ψ
2884    using "B◇" "Hypothetical Syllogism" by blast
2885qed
2886
2887AOT_theorem "BFs:1": α φ{α}  α φ{α}
2888proof -
2889  AOT_modally_strict {
2890    AOT_modally_strict {
2891      AOT_have α φ{α}  φ{α} for α by (fact AOT)
2892    }
2893    AOT_hence α φ{α}  φ{α} for α by (rule "RM◇")
2894    AOT_hence α φ{α}  α φ{α}
2895      using "B◇" "∀I" "→E" "→I" by metis
2896  }
2897  thus ?thesis using "derived-S5-rules:1" by blast
2898qed
2899lemmas "BF" = "BFs:1"
2900
2901AOT_theorem "BFs:2": α φ{α}  α φ{α}
2902proof -
2903  AOT_have α φ{α}  φ{α} for α using RM "cqt-orig:3" by metis
2904  thus ?thesis using  "cqt-orig:2"[THEN "→E"] "∀I" by metis
2905qed
2906lemmas "CBF" = "BFs:2"
2907
2908AOT_theorem "BFs:3": α φ{α}  α φ{α}
2909proof(rule "→I")
2910  AOT_modally_strict {
2911    AOT_have α ¬φ{α}  α ¬φ{α}
2912      using BF CBF "≡I" by blast
2913  } note θ = this
2914
2915  AOT_assume α φ{α}
2916  AOT_hence ¬¬(α φ{α})
2917    using "≡dfE" "conventions:5" by blast
2918  AOT_hence ¬α ¬φ{α}
2919    apply (AOT_subst α ¬φ{α} ¬(α φ{α}))
2920    using "≡dfI" "conventions:3" "conventions:4" "&I" "contraposition:2" "cqt-further:4"
2921          "df-rules-formulas[1]" "vdash-properties:1[2]" by blast
2922  AOT_hence ¬α ¬φ{α}
2923    apply (AOT_subst (reverse) α ¬φ{α} α ¬φ{α})
2924    using θ by blast
2925  AOT_hence ¬α ¬¬¬φ{α}
2926    apply - apply (AOT_subst_old_rev "λ τ. «¬φ{τ}»"  "λ τ. «¬¬¬φ{τ}»")
2927    by (simp add: "oth-class-taut:3:b")
2928  AOT_hence 0: α ¬¬φ{α}
2929    by (rule "conventions:4"[THEN "≡dfI"])
2930  AOT_show α φ{α}
2931    apply (AOT_subst_old "λ τ . «φ{τ}»" "λ τ . «¬¬φ{τ}»")
2932     apply (simp add: "conventions:5" "≡Df")
2933    using 0 by blast
2934qed
2935lemmas "BF◇" = "BFs:3"
2936
2937AOT_theorem "BFs:4": α φ{α}  α φ{α}
2938proof(rule "→I")
2939  AOT_assume α φ{α}
2940  AOT_hence ¬α ¬φ{α}
2941    using "conventions:4"[THEN "≡dfE"] by blast
2942  AOT_hence ¬α ¬φ{α}
2943    apply - apply (AOT_subst_old "λ τ . «¬φ{τ}»" "λ τ . «¬φ{τ}»")
2944    by (simp add: "KBasic2:1")
2945  moreover AOT_have α ¬φ{α}  α ¬φ{α}
2946    using "≡I" "BF" "CBF" by metis
2947  ultimately AOT_have 1: ¬α ¬φ{α}
2948    using "≡E"(3) by blast
2949  AOT_show α φ{α}
2950    apply (rule "conventions:5"[THEN "≡dfI"])
2951    apply (AOT_subst α φ{α} ¬α ¬φ{α})
2952     apply (simp add: "conventions:4" "≡Df")
2953    apply (AOT_subst ¬¬α ¬φ{α} α ¬φ{α})
2954    by (auto simp: 1 "≡I" "useful-tautologies:1" "useful-tautologies:2")
2955qed
2956lemmas "CBF◇" = "BFs:4"
2957
2958AOT_theorem "sign-S5-thm:1": α φ{α}  α φ{α}
2959proof(rule "→I")
2960  AOT_assume α φ{α}
2961  then AOT_obtain α where φ{α} using "∃E" by metis
2962  moreover AOT_have α
2963    by (simp add: "ex:1:a" "rule-ui:2[const_var]" RN)
2964  moreover AOT_have φ{τ}, τ  α φ{α} for τ
2965  proof -
2966    AOT_have φ{τ}, τ  α φ{α} using "existential:1" by blast
2967    AOT_thus φ{τ}, τ  α φ{α}
2968      using "RN[prem]"[where Γ="{φ τ, «τ»}", simplified] by blast
2969  qed
2970  ultimately AOT_show α φ{α} by blast
2971qed
2972lemmas Buridan = "sign-S5-thm:1"
2973
2974AOT_theorem "sign-S5-thm:2": α φ{α}  α φ{α}
2975proof -
2976  AOT_have α (α φ{α}  φ{α})
2977    by (simp add: "RM◇" "cqt-orig:3" "∀I")
2978  AOT_thus α φ{α}  α φ{α}
2979    using "∀E"(4) "∀I" "→E" "→I" by metis
2980qed
2981lemmas "Buridan◇" = "sign-S5-thm:2"
2982
2983AOT_theorem "sign-S5-thm:3": α (φ{α} & ψ{α})  (α φ{α} & α ψ{α})
2984  apply (rule "RM:2")
2985  by (metis (no_types, lifting) "instantiation" "&I" "&E"(1)
2986                                "&E"(2) "deduction-theorem" "existential:2[const_var]")
2987
2988AOT_theorem "sign-S5-thm:4": α (φ{α} & ψ{α})  α φ{α}
2989  apply (rule "RM:2")
2990  by (meson "instantiation" "&E"(1) "deduction-theorem" "existential:2[const_var]")
2991
2992AOT_theorem "sign-S5-thm:5": (α (φ{α}  ψ{α}) & α (ψ{α}  χ{α}))  α (φ{α}  χ{α})
2993proof -
2994  {
2995    fix φ' ψ' χ'
2996    AOT_assume  φ' & ψ'  χ'
2997    AOT_hence φ' & ψ'  χ'
2998      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2999      using "&E" "&I" "→E" "→I" by metis
3000  } note R = this
3001  show ?thesis by (rule R; fact AOT)
3002qed
3003
3004AOT_theorem "sign-S5-thm:6": (α (φ{α}  ψ{α}) & α(ψ{α}  χ{α}))  α(φ{α}  χ{α})
3005proof -
3006  {
3007    fix φ' ψ' χ'
3008    AOT_assume  φ' & ψ'  χ'
3009    AOT_hence φ' & ψ'  χ'
3010      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
3011      using "&E" "&I" "→E" "→I" by metis
3012  } note R = this
3013  show ?thesis by (rule R; fact AOT)
3014qed
3015
3016AOT_theorem "exist-nec2:1": τ  τ
3017  using "B◇" "RM◇" "Hypothetical Syllogism" "exist-nec" by blast
3018
3019AOT_theorem "exists-nec2:2": τ  τ
3020  by (meson "Act-Sub:3" "Hypothetical Syllogism" "exist-nec" "exist-nec2:1" "≡I" "nec-imp-act")
3021
3022AOT_theorem "exists-nec2:3": ¬τ  ¬τ
3023  using "KBasic2:1" "deduction-theorem" "exist-nec2:1" "≡E"(2) "modus-tollens:1" by blast
3024
3025AOT_theorem "exists-nec2:4": ¬τ  ¬τ
3026  by (metis "Act-Sub:3" "KBasic:12" "deduction-theorem" "exist-nec" "exists-nec2:3" "≡I" "≡E"(4) "nec-imp-act" "reductio-aa:1")
3027
3028AOT_theorem "id-nec2:1": α = β  α = β
3029  using "B◇" "RM◇" "Hypothetical Syllogism" "id-nec:1" by blast
3030
3031AOT_theorem "id-nec2:2": α  β  α  β
3032  apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
3033  using "KBasic2:1" "deduction-theorem" "id-nec2:1" "≡E"(2) "modus-tollens:1" by blast
3034
3035AOT_theorem "id-nec2:3": α  β  α  β
3036  apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
3037  by (metis "KBasic:11" "deduction-theorem" "id-nec:2" "≡E"(3) "reductio-aa:2" "vdash-properties:6")
3038
3039AOT_theorem "id-nec2:4": α = β  α = β
3040  using "Hypothetical Syllogism" "id-nec2:1" "id-nec:1" by blast
3041
3042AOT_theorem "id-nec2:5": α  β  α  β
3043  using "id-nec2:3" "id-nec2:2" "→I" "→E" by metis
3044
3045AOT_theorem "sc-eq-box-box:1": (φ  φ)  (φ  φ)
3046  apply (rule "≡I"; rule "→I")
3047  using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:10" apply blast
3048  by (metis "KBasic2:1" "KBasic:1" "KBasic:2" "S5Basic:13" "≡E"(2) "raa-cor:5" "vdash-properties:6")
3049
3050AOT_theorem "sc-eq-box-box:2": ((φ  φ)  (φ  φ))  (φ  φ)
3051  by (metis "Act-Sub:3" "KBasic:13" "5◇" "∨E"(2) "deduction-theorem" "≡I" "nec-imp-act" "raa-cor:2" "vdash-properties:10")
3052
3053AOT_theorem "sc-eq-box-box:3": (φ  φ)  (¬φ  ¬φ)
3054proof (rule "→I"; rule "≡I"; rule "→I")
3055  AOT_assume (φ  φ)
3056  AOT_hence φ  φ using "sc-eq-box-box:1" "≡E" by blast
3057  moreover AOT_assume ¬φ
3058  ultimately AOT_have ¬φ
3059    using "modus-tollens:1" by blast
3060  AOT_thus ¬φ
3061    using "KBasic2:1" "≡E"(2) by blast
3062next
3063  AOT_assume (φ  φ)
3064  moreover AOT_assume ¬φ
3065  ultimately AOT_show ¬φ
3066    using "modus-tollens:1" "qml:2" "vdash-properties:10" "vdash-properties:1[2]" by blast
3067qed
3068
3069AOT_theorem "sc-eq-box-box:4": ((φ  φ) & (ψ  ψ))  ((φ  ψ)  (φ  ψ))
3070proof(rule "→I"; rule "→I")
3071  AOT_assume θ: (φ  φ) & (ψ  ψ)
3072  AOT_assume ξ: φ  ψ
3073  AOT_hence (φ & ψ)  (¬φ & ¬ψ)
3074    using "≡E"(4) "oth-class-taut:4:g" "raa-cor:3" by blast
3075  moreover {
3076    AOT_assume φ & ψ
3077    AOT_hence (φ  ψ)
3078      using "KBasic:3" "KBasic:8" "≡E"(2) "vdash-properties:10" by blast
3079  }
3080  moreover {
3081    AOT_assume ¬φ & ¬ψ
3082    moreover AOT_have ¬φ  ¬φ and ¬ψ  ¬ψ
3083      using θ "Conjunction Simplification"(1) "Conjunction Simplification"(2) "sc-eq-box-box:3" "vdash-properties:10" by metis+
3084    ultimately AOT_have ¬φ & ¬ψ
3085      by (metis "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "≡E"(4) "modus-tollens:1" "raa-cor:3")
3086    AOT_hence (φ  ψ)
3087      using "KBasic:3" "KBasic:9" "≡E"(2) "vdash-properties:10" by blast
3088  }
3089  ultimately AOT_show (φ  ψ)
3090    using "∨E"(2) "reductio-aa:1" by blast
3091qed
3092
3093AOT_theorem "sc-eq-box-box:5": ((φ  φ) & (ψ  ψ))  ((φ  ψ)  (φ  ψ))
3094proof (rule "→I")
3095  AOT_assume ((φ  φ) & (ψ  ψ))
3096  AOT_hence ((φ  φ) & (ψ  ψ))
3097    using 4[THEN "→E"] "&E" "&I" "KBasic:3" "≡E"(2) by metis
3098  moreover AOT_have ((φ  φ) & (ψ  ψ))  ((φ  ψ)  (φ  ψ))
3099  proof (rule RM; rule "→I"; rule "→I")
3100    AOT_modally_strict {
3101      AOT_assume A: ((φ  φ) & (ψ  ψ))
3102      AOT_hence φ  φ and ψ  ψ
3103        using "&E" "qml:2"[axiom_inst] "→E" by blast+
3104      moreover AOT_assume φ  ψ
3105      ultimately AOT_have φ  ψ
3106        using "→E" "qml:2"[axiom_inst] "≡E" "≡I" by meson
3107      moreover AOT_have (φ  ψ)  (φ  ψ)
3108        using A "sc-eq-box-box:4" "→E" by blast
3109      ultimately AOT_show (φ  ψ) using "→E" by blast
3110    }
3111  qed
3112  ultimately AOT_show ((φ  ψ)  (φ  ψ)) using "→E" by blast
3113qed
3114
3115AOT_theorem "sc-eq-box-box:6": (φ  φ)  ((φ  ψ)  (φ  ψ))
3116proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3117  AOT_assume ¬(φ  ψ)
3118  AOT_hence ¬(φ  ψ) by (metis "KBasic:11" "≡E"(1))
3119  AOT_hence (φ & ¬ψ)
3120    by (AOT_subst φ & ¬ψ ¬(φ  ψ))
3121       (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3122  AOT_hence φ and 2: ¬ψ using "KBasic2:3"[THEN "→E"] "&E" by blast+
3123  moreover AOT_assume (φ  φ)
3124  ultimately AOT_have φ by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3125  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3126  moreover AOT_assume φ  ψ
3127  ultimately AOT_have ψ using "→E" by blast
3128  moreover AOT_have ¬ψ using 2 "KBasic:12" "¬¬I" "intro-elim:3:d" by blast
3129  ultimately AOT_show ψ & ¬ψ using "&I" by blast
3130qed
3131
3132AOT_theorem "sc-eq-box-box:7": (φ  φ)  ((φ  𝒜ψ)  𝒜(φ  ψ))
3133proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3134  AOT_assume ¬𝒜(φ  ψ)
3135  AOT_hence 𝒜¬(φ  ψ) by (metis "Act-Basic:1" "∨E"(2))
3136  AOT_hence 𝒜(φ & ¬ψ)
3137    by (AOT_subst φ & ¬ψ ¬(φ  ψ))
3138       (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3139  AOT_hence 𝒜φ and 2: 𝒜¬ψ using "Act-Basic:2"[THEN "≡E"(1)] "&E" by blast+
3140  AOT_hence φ by (metis "Act-Sub:3" "→E")
3141  moreover AOT_assume (φ  φ)
3142  ultimately AOT_have φ by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3143  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3144  moreover AOT_assume φ  𝒜ψ
3145  ultimately AOT_have 𝒜ψ using "→E" by blast
3146  moreover AOT_have ¬𝒜ψ using 2 by (meson "Act-Sub:1" "≡E"(4) "raa-cor:3")
3147  ultimately AOT_show 𝒜ψ & ¬𝒜ψ using "&I" by blast
3148qed
3149
3150AOT_theorem "sc-eq-fur:1": 𝒜φ  𝒜φ
3151  using "Act-Basic:6" "Act-Sub:4" "≡E"(6) by blast
3152
3153AOT_theorem "sc-eq-fur:2": (φ  φ)  (𝒜φ  φ)
3154  by (metis "B◇" "Act-Sub:3" "KBasic:13" "T◇" "Hypothetical Syllogism" "deduction-theorem" "≡I" "nec-imp-act")
3155
3156AOT_theorem "sc-eq-fur:3": x (φ{x}  φ{x})  (∃!x φ{x}  ιx φ{x})
3157proof (rule "→I"; rule "→I")
3158  AOT_assume x (φ{x}  φ{x})
3159  AOT_hence A: x (φ{x}  φ{x}) using CBF "→E" by blast
3160  AOT_assume ∃!x φ{x}
3161  then AOT_obtain a where a_def: φ{a} & y (φ{y}  y = a)
3162    using "∃E"[rotated 1, OF "uniqueness:1"[THEN "≡dfE"]] by blast
3163  moreover AOT_have φ{a} using calculation A "∀E"(2) "qml:2"[axiom_inst] "→E" "&E"(1) by blast
3164  AOT_hence 𝒜φ{a} using "nec-imp-act" "vdash-properties:6" by blast
3165  moreover AOT_have y (𝒜φ{y}  y = a)
3166  proof (rule "∀I"; rule "→I")
3167    fix b
3168    AOT_assume 𝒜φ{b}
3169    AOT_hence φ{b}
3170      using "Act-Sub:3" "vdash-properties:6" by blast
3171    moreover {
3172      AOT_have (φ{b}  φ{b})
3173        using A "∀E"(2) by blast
3174      AOT_hence φ{b}  φ{b}
3175        using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:6" by blast
3176    }
3177    ultimately AOT_have φ{b} using "→E" by blast
3178    AOT_hence φ{b} using "qml:2"[axiom_inst] "→E" by blast
3179    AOT_thus b = a
3180      using a_def[THEN "&E"(2)] "∀E"(2) "→E" by blast
3181  qed
3182  ultimately AOT_have 𝒜φ{a} & y (𝒜φ{y}  y = a)
3183    using "&I" by blast
3184  AOT_hence x (𝒜φ{x} & y (𝒜φ{y}  y = x)) using "∃I" by fast
3185  AOT_hence ∃!x 𝒜φ{x} using "uniqueness:1"[THEN "≡dfI"] by fast
3186  AOT_thus ιx φ{x}
3187    using "actual-desc:1"[THEN "≡E"(2)] by blast
3188qed
3189
3190AOT_theorem "sc-eq-fur:4": x (φ{x}  φ{x})  (x = ιx φ{x}  (φ{x} & z (φ{z}  z = x)))
3191proof (rule "→I")
3192  AOT_assume x (φ{x}  φ{x})
3193  AOT_hence x (φ{x}  φ{x}) using CBF "→E" by blast
3194  AOT_hence A: 𝒜φ{α}  φ{α} for α using "sc-eq-fur:2" "∀E" "→E" by fast
3195  AOT_show x = ιx φ{x}  (φ{x} & z (φ{z}  z = x))
3196  proof (rule "≡I"; rule "→I")
3197    AOT_assume x = ιx φ{x}
3198    AOT_hence B: 𝒜φ{x} & z (𝒜φ{z}  z = x)
3199      using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
3200    AOT_show φ{x} & z (φ{z}  z = x)
3201    proof (rule "&I"; (rule "∀I"; rule "→I")?)
3202      AOT_show φ{x} using A B[THEN "&E"(1)] "≡E"(1) by blast
3203    next
3204      AOT_show z = x if φ{z} for z
3205        using that B[THEN "&E"(2)] "∀E"(2) "→E" A[THEN "≡E"(2)] by blast
3206    qed
3207  next
3208    AOT_assume B: φ{x} & z (φ{z}  z = x)
3209    AOT_have 𝒜φ{x} & z (𝒜φ{z}  z = x)
3210    proof(rule "&I"; (rule "∀I"; rule "→I")?)
3211      AOT_show 𝒜φ{x} using B[THEN "&E"(1)] A[THEN "≡E"(2)] by blast
3212    next
3213      AOT_show b = x if 𝒜φ{b} for b
3214        using that A[THEN "≡E"(1)] B[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] by blast
3215    qed
3216    AOT_thus x = ιx φ{x}
3217      using "nec-hintikka-scheme"[THEN "≡E"(2)] by blast
3218  qed
3219qed
3220
3221AOT_theorem "id-act:1": α = β  𝒜α = β
3222  by (meson "Act-Sub:3" "Hypothetical Syllogism" "id-nec2:1" "id-nec:2" "≡I" "nec-imp-act")
3223
3224AOT_theorem "id-act:2": α  β  𝒜α  β
3225proof (AOT_subst α  β ¬(α = β))
3226  AOT_modally_strict {
3227    AOT_show α  β  ¬(α = β)
3228      by (simp add: "=-infix" "≡Df")
3229  }
3230next
3231  AOT_show ¬(α = β)  𝒜¬(α = β)
3232  proof (safe intro!: "≡I" "→I")
3233    AOT_assume ¬α = β
3234    AOT_hence ¬𝒜α = β using "id-act:1" "≡E"(3) by blast
3235    AOT_thus 𝒜¬α = β
3236      using "¬¬E" "Act-Sub:1" "≡E"(3) by blast
3237  next
3238    AOT_assume 𝒜¬α = β
3239    AOT_hence ¬𝒜α = β
3240      using "¬¬I" "Act-Sub:1" "≡E"(4) by blast
3241    AOT_thus ¬α = β
3242      using "id-act:1" "≡E"(4) by blast
3243  qed
3244qed
3245
3246AOT_theorem "A-Exists:1": 𝒜∃!α φ{α}  ∃!α 𝒜φ{α}
3247proof -
3248  AOT_have 𝒜∃!α φ{α}  𝒜αβ (φ{β}  β = α)
3249    by (AOT_subst_using subst: "uniqueness:2")
3250       (simp add: "oth-class-taut:3:a")
3251  also AOT_have   α 𝒜β (φ{β}  β = α)
3252    by (simp add: "Act-Basic:10")
3253  also AOT_have   αβ 𝒜(φ{β}  β = α)
3254    by (AOT_subst_old "λ τ . «𝒜β (φ{β}  β = τ)»" "λ τ . «β 𝒜(φ{β}  β = τ)»")
3255       (auto simp: "logic-actual-nec:3" "vdash-properties:1[2]" "oth-class-taut:3:a")
3256  also AOT_have   αβ (𝒜φ{β}  𝒜β = α)
3257    by (AOT_subst_old_rev "λ τ τ' . «𝒜(φ{τ'}  τ' = τ)»" "λ τ τ'. «𝒜φ{τ'}  𝒜τ' = τ»")
3258       (auto simp: "Act-Basic:5" "cqt-further:7")
3259  also AOT_have   αβ (𝒜φ{β}  β = α)
3260    apply (AOT_subst_old "λ τ τ' :: 'a . «𝒜τ' = τ»" "λ τ τ'. «τ' = τ»")
3261     apply (meson "id-act:1" "≡E"(6) "oth-class-taut:3:a")
3262    by (simp add: "cqt-further:7")
3263  also AOT_have ...  ∃!α 𝒜φ{α}
3264    using "uniqueness:2" "Commutativity of ≡"[THEN "≡E"(1)] by fast
3265  finally show ?thesis .
3266qed
3267
3268AOT_theorem "A-Exists:2": ιx φ{x}  𝒜∃!x φ{x}
3269  by (AOT_subst_using subst: "A-Exists:1")
3270     (simp add: "actual-desc:1")
3271
3272AOT_theorem "id-act-desc:1": ιx (x = y)
3273proof(rule "existence:1"[THEN "≡dfI"]; rule "∃I")
3274  AOT_show x E!x  E!x]ιx (x = y)
3275  proof (rule "russell-axiom[exe,1].nec-russell-axiom"[THEN "≡E"(2)]; rule "∃I"; (rule "&I")+)
3276    AOT_show 𝒜y = y by (simp add: "RA[2]" "id-eq:1")
3277  next
3278    AOT_show z (𝒜z = y  z = y)
3279      apply (rule "∀I")
3280      using "id-act:1"[THEN "≡E"(2)] "→I" by blast
3281  next
3282    AOT_show x E!x  E!x]y
3283    proof (rule "lambda-predicates:2"[axiom_inst, THEN "→E", THEN "≡E"(2)])
3284      AOT_show x E!x  E!x]
3285        by "cqt:2[lambda]"
3286    next
3287      AOT_show E!y  E!y 
3288        by (simp add: "if-p-then-p")
3289    qed
3290  qed
3291next
3292  AOT_show x E!x  E!x]
3293    by "cqt:2[lambda]"
3294qed
3295
3296AOT_theorem "id-act-desc:2": y = ιx (x = y)
3297  by (rule descriptions[axiom_inst, THEN "≡E"(2)]; rule "∀I"; rule "id-act:1"[symmetric])
3298
3299AOT_theorem "pre-en-eq:1[1]": x1[F]  x1[F]
3300  by (simp add: encoding "vdash-properties:1[2]")
3301
3302AOT_theorem "pre-en-eq:1[2]": x1x2[F]  x1x2[F]
3303proof (rule "→I")
3304  AOT_assume x1x2[F]
3305  AOT_hence x1y [F]yx2] and x2y [F]x1y]
3306    using "nary-encoding[2]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3307  moreover AOT_have y [F]yx2] by "cqt:2[lambda]"
3308  moreover AOT_have y [F]x1y] by "cqt:2[lambda]"
3309  ultimately AOT_have x1y [F]yx2] and x2y [F]x1y]
3310    using encoding[axiom_inst, unvarify F] "→E" "&I" by blast+
3311  note A = this
3312  AOT_hence (x1y [F]yx2] & x2y [F]x1y])
3313    using "KBasic:3"[THEN "≡E"(2)] "&I" by blast
3314  AOT_thus x1x2[F]
3315    by (rule "nary-encoding[2]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3316qed
3317
3318AOT_theorem "pre-en-eq:1[3]": x1x2x3[F]  x1x2x3[F]
3319proof (rule "→I")
3320  AOT_assume x1x2x3[F]
3321  AOT_hence x1y [F]yx2x3] and x2y [F]x1yx3] and x3y [F]x1x2y]
3322    using "nary-encoding[3]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3323  moreover AOT_have y [F]yx2x3] by "cqt:2[lambda]"
3324  moreover AOT_have y [F]x1yx3] by "cqt:2[lambda]"
3325  moreover AOT_have y [F]x1x2y] by "cqt:2[lambda]"
3326  ultimately AOT_have x1y [F]yx2x3] and x2y [F]x1yx3] and x3y [F]x1x2y]
3327    using encoding[axiom_inst, unvarify F] "→E" by blast+
3328  note A = this
3329  AOT_have B: (x1y [F]yx2x3] & x2y [F]x1yx3] & x3y [F]x1x2y])
3330    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3331  AOT_thus x1x2x3[F]
3332    by (rule "nary-encoding[3]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3333qed
3334
3335AOT_theorem "pre-en-eq:1[4]": x1x2x3x4[F]  x1x2x3x4[F]
3336proof (rule "→I")
3337  AOT_assume x1x2x3x4[F]
3338  AOT_hence x1y [F]yx2x3x4] and x2y [F]x1yx3x4] and x3y [F]x1x2yx4] and  x4y [F]x1x2x3y]
3339    using "nary-encoding[4]"[axiom_inst, THEN "≡E"(1)] "&E" by metis+
3340  moreover AOT_have y [F]yx2x3x4] by "cqt:2[lambda]"
3341  moreover AOT_have y [F]x1yx3x4] by "cqt:2[lambda]"
3342  moreover AOT_have y [F]x1x2yx4] by "cqt:2[lambda]"
3343  moreover AOT_have y [F]x1x2x3y] by "cqt:2[lambda]"
3344  ultimately AOT_have x1y [F]yx2x3x4] and x2y [F]x1yx3x4] and x3y [F]x1x2yx4] and x4y [F]x1x2x3y]
3345    using "→E" encoding[axiom_inst, unvarify F] by blast+
3346  note A = this
3347  AOT_have B: (x1y [F]yx2x3x4] & x2y [F]x1yx3x4] & x3y [F]x1x2yx4] & x4y [F]x1x2x3y])
3348    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3349  AOT_thus x1x2x3x4[F]
3350    by (rule "nary-encoding[4]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3351qed
3352
3353AOT_theorem "pre-en-eq:2[1]": ¬x1[F]  ¬x1[F]
3354proof (rule "→I"; rule "raa-cor:1")
3355  AOT_assume ¬¬x1[F]
3356  AOT_hence x1[F]
3357    by (rule "conventions:5"[THEN "≡dfI"])
3358  AOT_hence x1[F]
3359    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[1]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3360  moreover AOT_assume ¬x1[F]
3361  ultimately AOT_show x1[F] & ¬x1[F] by (rule "&I")
3362qed
3363AOT_theorem "pre-en-eq:2[2]": ¬x1x2[F]  ¬x1x2[F]
3364proof (rule "→I"; rule "raa-cor:1")
3365  AOT_assume ¬¬x1x2[F]
3366  AOT_hence x1x2[F]
3367    by (rule "conventions:5"[THEN "≡dfI"])
3368  AOT_hence x1x2[F]
3369    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[2]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3370  moreover AOT_assume ¬x1x2[F]
3371  ultimately AOT_show x1x2[F] & ¬x1x2[F] by (rule "&I")
3372qed
3373
3374AOT_theorem "pre-en-eq:2[3]": ¬x1x2x3[F]  ¬x1x2x3[F]
3375proof (rule "→I"; rule "raa-cor:1")
3376  AOT_assume ¬¬x1x2x3[F]
3377  AOT_hence x1x2x3[F]
3378    by (rule "conventions:5"[THEN "≡dfI"])
3379  AOT_hence x1x2x3[F]
3380    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[3]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3381  moreover AOT_assume ¬x1x2x3[F]
3382  ultimately AOT_show x1x2x3[F] & ¬x1x2x3[F] by (rule "&I")
3383qed
3384
3385AOT_theorem "pre-en-eq:2[4]": ¬x1x2x3x4[F]  ¬x1x2x3x4[F]
3386proof (rule "→I"; rule "raa-cor:1")
3387  AOT_assume ¬¬x1x2x3x4[F]
3388  AOT_hence x1x2x3x4[F]
3389    by (rule "conventions:5"[THEN "≡dfI"])
3390  AOT_hence x1x2x3x4[F]
3391    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[4]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3392  moreover AOT_assume ¬x1x2x3x4[F]
3393  ultimately AOT_show x1x2x3x4[F] & ¬x1x2x3x4[F] by (rule "&I")
3394qed
3395
3396AOT_theorem "en-eq:1[1]": x1[F]  x1[F]
3397  using "pre-en-eq:1[1]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3398AOT_theorem "en-eq:1[2]": x1x2[F]  x1x2[F]
3399  using "pre-en-eq:1[2]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3400AOT_theorem "en-eq:1[3]": x1x2x3[F]  x1x2x3[F]
3401  using "pre-en-eq:1[3]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3402AOT_theorem "en-eq:1[4]": x1x2x3x4[F]  x1x2x3x4[F]
3403  using "pre-en-eq:1[4]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3404
3405AOT_theorem "en-eq:2[1]": x1[F]  x1[F]
3406  by (simp add: "≡I" "pre-en-eq:1[1]" "qml:2"[axiom_inst])
3407AOT_theorem "en-eq:2[2]": x1x2[F]  x1x2[F]
3408  by (simp add: "≡I" "pre-en-eq:1[2]" "qml:2"[axiom_inst])
3409AOT_theorem "en-eq:2[3]": x1x2x3[F]  x1x2x3[F]
3410  by (simp add: "≡I" "pre-en-eq:1[3]" "qml:2"[axiom_inst])
3411AOT_theorem "en-eq:2[4]": x1x2x3x4[F]  x1x2x3x4[F]
3412  by (simp add: "≡I" "pre-en-eq:1[4]" "qml:2"[axiom_inst])
3413
3414AOT_theorem "en-eq:3[1]": x1[F]  x1[F]
3415  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[1]"] "≡I" by blast
3416AOT_theorem "en-eq:3[2]": x1x2[F]  x1x2[F]
3417  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[2]"] "≡I" by blast
3418AOT_theorem "en-eq:3[3]": x1x2x3[F]  x1x2x3[F]
3419  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[3]"] "≡I" by blast
3420AOT_theorem "en-eq:3[4]": x1x2x3x4[F]  x1x2x3x4[F]
3421  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[4]"] "≡I" by blast
3422
3423AOT_theorem "en-eq:4[1]": (x1[F]  y1[G])  (x1[F]  y1[G])
3424  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3425  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[1]" by blast+
3426AOT_theorem "en-eq:4[2]": (x1x2[F]  y1y2[G])  (x1x2[F]  y1y2[G])
3427  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3428  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[2]" by blast+
3429AOT_theorem "en-eq:4[3]": (x1x2x3[F]  y1y2y3[G])  (x1x2x3[F]  y1y2y3[G])
3430  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3431  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[3]" by blast+
3432AOT_theorem "en-eq:4[4]": (x1x2x3x4[F]  y1y2y3y4[G])  (x1x2x3x4[F]  y1y2y3y4[G])
3433  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3434  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[4]" by blast+
3435
3436AOT_theorem "en-eq:5[1]": (x1[F]  y1[G])  (x1[F]  y1[G])
3437  apply (rule "≡I"; rule "→I")
3438  using "en-eq:4[1]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3439  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3440        "&I"[OF "pre-en-eq:1[1]"[THEN RN], OF "pre-en-eq:1[1]"[THEN RN]] by blast
3441AOT_theorem "en-eq:5[2]": (x1x2[F]  y1y2[G])  (x1x2[F]  y1y2[G])
3442  apply (rule "≡I"; rule "→I")
3443  using "en-eq:4[2]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3444  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3445        "&I"[OF "pre-en-eq:1[2]"[THEN RN], OF "pre-en-eq:1[2]"[THEN RN]] by blast
3446AOT_theorem "en-eq:5[3]": (x1x2x3[F]  y1y2y3[G])  (x1x2x3[F]  y1y2y3[G])
3447  apply (rule "≡I"; rule "→I")
3448  using "en-eq:4[3]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3449  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3450        "&I"[OF "pre-en-eq:1[3]"[THEN RN], OF "pre-en-eq:1[3]"[THEN RN]] by blast
3451AOT_theorem "en-eq:5[4]": (x1x2x3x4[F]  y1y2y3y4[G])  (x1x2x3x4[F]  y1y2y3y4[G])
3452  apply (rule "≡I"; rule "→I")
3453  using "en-eq:4[4]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3454  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3455        "&I"[OF "pre-en-eq:1[4]"[THEN RN], OF "pre-en-eq:1[4]"[THEN RN]] by blast
3456
3457AOT_theorem "en-eq:6[1]": (x1[F]  y1[G])  (x1[F]  y1[G])
3458  using "en-eq:5[1]"[symmetric] "en-eq:4[1]" "≡E"(5) by fast
3459AOT_theorem "en-eq:6[2]": (x1x2[F]  y1y2[G])  (x1x2[F]  y1y2[G])
3460  using "en-eq:5[2]"[symmetric] "en-eq:4[2]" "≡E"(5) by fast
3461AOT_theorem "en-eq:6[3]": (x1x2x3[F]  y1y2y3[G])  (x1x2x3[F]  y1y2y3[G])
3462  using "en-eq:5[3]"[symmetric] "en-eq:4[3]" "≡E"(5) by fast
3463AOT_theorem "en-eq:6[4]": (x1x2x3x4[F]  y1y2y3y4[G])  (x1x2x3x4[F]  y1y2y3y4[G])
3464  using "en-eq:5[4]"[symmetric] "en-eq:4[4]" "≡E"(5) by fast
3465
3466AOT_theorem "en-eq:7[1]": ¬x1[F]  ¬x1[F]
3467  using "pre-en-eq:2[1]" "qml:2"[axiom_inst] "≡I" by blast
3468AOT_theorem "en-eq:7[2]": ¬x1x2[F]  ¬x1x2[F]
3469  using "pre-en-eq:2[2]" "qml:2"[axiom_inst] "≡I" by blast
3470AOT_theorem "en-eq:7[3]": ¬x1x2x3[F]  ¬x1x2x3[F]
3471  using "pre-en-eq:2[3]" "qml:2"[axiom_inst] "≡I" by blast
3472AOT_theorem "en-eq:7[4]": ¬x1x2x3x4[F]  ¬x1x2x3x4[F]
3473  using "pre-en-eq:2[4]" "qml:2"[axiom_inst] "≡I" by blast
3474
3475AOT_theorem "en-eq:8[1]": ¬x1[F]  ¬x1[F]
3476  using "en-eq:2[1]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3477AOT_theorem "en-eq:8[2]": ¬x1x2[F]  ¬x1x2[F]
3478  using "en-eq:2[2]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3479AOT_theorem "en-eq:8[3]": ¬x1x2x3[F]  ¬x1x2x3[F]
3480  using "en-eq:2[3]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3481AOT_theorem "en-eq:8[4]": ¬x1x2x3x4[F]  ¬x1x2x3x4[F]
3482  using "en-eq:2[4]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3483
3484AOT_theorem "en-eq:9[1]": ¬x1[F]  ¬x1[F]
3485  using "en-eq:7[1]" "en-eq:8[1]" "≡E"(5) by blast
3486AOT_theorem "en-eq:9[2]": ¬x1x2[F]  ¬x1x2[F]
3487  using "en-eq:7[2]" "en-eq:8[2]" "≡E"(5) by blast
3488AOT_theorem "en-eq:9[3]": ¬x1x2x3[F]  ¬x1x2x3[F]
3489  using "en-eq:7[3]" "en-eq:8[3]" "≡E"(5) by blast
3490AOT_theorem "en-eq:9[4]": ¬x1x2x3x4[F]  ¬x1x2x3x4[F]
3491  using "en-eq:7[4]" "en-eq:8[4]" "≡E"(5) by blast
3492
3493AOT_theorem "en-eq:10[1]": 𝒜x1[F]  x1[F]
3494  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[1]" "pre-en-eq:1[1]")
3495AOT_theorem "en-eq:10[2]": 𝒜x1x2[F]  x1x2[F]
3496  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[2]" "pre-en-eq:1[2]")
3497AOT_theorem "en-eq:10[3]": 𝒜x1x2x3[F]  x1x2x3[F]
3498  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[3]" "pre-en-eq:1[3]")
3499AOT_theorem "en-eq:10[4]": 𝒜x1x2x3x4[F]  x1x2x3x4[F]
3500  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[4]" "pre-en-eq:1[4]")
3501
3502AOT_theorem "oa-facts:1": O!x  O!x
3503proof(rule "→I")
3504  AOT_modally_strict {
3505    AOT_have x E!x]x  E!x
3506      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3507  } note θ = this
3508  AOT_assume O!x
3509  AOT_hence x E!x]x
3510    by (rule "=dfE"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3511  AOT_hence E!x using θ[THEN "≡E"(1)] by blast
3512  AOT_hence 0: E!x using "qml:3"[axiom_inst, THEN "→E"] by blast
3513  AOT_have x E!x]x
3514    by (AOT_subst_using subst: θ) (simp add: 0)
3515  AOT_thus O!x
3516    by (rule "=dfI"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3517qed
3518
3519AOT_theorem "oa-facts:2": A!x  A!x
3520proof(rule "→I")
3521  AOT_modally_strict {
3522    AOT_have x ¬E!x]x  ¬E!x
3523      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3524  } note θ = this
3525  AOT_assume A!x
3526  AOT_hence x ¬E!x]x
3527    by (rule "=dfE"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3528  AOT_hence ¬E!x using θ[THEN "≡E"(1)] by blast
3529  AOT_hence ¬E!x using "KBasic2:1"[THEN "≡E"(2)] by blast
3530  AOT_hence 0: ¬E!x using "4"[THEN "→E"] by blast
3531  AOT_have 1: ¬E!x
3532    apply (AOT_subst ¬E!x ¬E!x)
3533    using "KBasic2:1"[symmetric] apply blast
3534    using 0 by blast
3535  AOT_have x ¬E!x]x
3536    by (AOT_subst_using subst: θ) (simp add: 1)
3537  AOT_thus A!x
3538    by (rule "=dfI"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3539qed
3540
3541AOT_theorem "oa-facts:3": O!x  O!x
3542  using "oa-facts:1" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3543AOT_theorem "oa-facts:4": A!x  A!x
3544  using "oa-facts:2" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3545
3546AOT_theorem "oa-facts:5": O!x  O!x
3547  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3548
3549AOT_theorem "oa-facts:6": A!x  A!x
3550  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3551
3552AOT_theorem "oa-facts:7": O!x  𝒜O!x
3553  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3554
3555AOT_theorem "oa-facts:8": A!x  𝒜A!x
3556  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3557
3558AOT_theorem "beta-C-meta": μ1...μn φ{μ1...μn, ν1...νn}]  (μ1...μn φ{μ1...μn, ν1...νn}]ν1...νn  φ{ν1...νn, ν1...νn})
3559  using "lambda-predicates:2"[axiom_inst] by blast
3560
3561AOT_theorem "beta-C-cor:1": (ν1...∀νn(μ1...μn φ{μ1...μn, ν1...νn}]))  ν1...∀νn (μ1...μn φ{μ1...μn, ν1...νn}]ν1...νn  φ{ν1...νn, ν1...νn})
3562  apply (rule "cqt-basic:14"[where 'a='a, THEN "→E"])
3563  using "beta-C-meta" "∀I" by fast
3564
3565AOT_theorem "beta-C-cor:2": μ1...μn φ{μ1...μn}]  ν1...∀νn (μ1...μn φ{μ1...μn}]ν1...νn  φ{ν1...νn})
3566  apply (rule "→I"; rule "∀I")
3567  using "beta-C-meta"[THEN "→E"] by fast
3568
3569(* TODO: syntax + double-check if this is really a faithful representation *)
3570theorem "beta-C-cor:3": assumes ν1νn. AOT_instance_of_cqt_2 (φ (AOT_term_of_var ν1νn))
3571  shows [v  ν1...∀νn (μ1...μn φ{ν1...νn,μ1...μn}]ν1...νn  φ{ν1...νn,ν1...νn})]
3572  using "cqt:2[lambda]"[axiom_inst, OF assms] "beta-C-cor:1"[THEN "→E"] "∀I" by fast
3573
3574AOT_theorem "betaC:1:a": μ1...μn φ{μ1...μn}]κ1...κn  φ{κ1...κn}
3575proof -
3576  AOT_modally_strict {
3577    AOT_assume μ1...μn φ{μ1...μn}]κ1...κn
3578    moreover AOT_have μ1...μn φ{μ1...μn}] and κ1...κn
3579      using calculation "cqt:5:a"[axiom_inst, THEN "→E"] "&E" by blast+
3580    ultimately AOT_show φ{κ1...κn}
3581      using "beta-C-cor:2"[THEN "→E", THEN "∀E"(1), THEN "≡E"(1)] by blast
3582  }
3583qed
3584
3585AOT_theorem "betaC:1:b": ¬φ{κ1...κn}  ¬μ1...μn φ{μ1...μn}]κ1...κn
3586  using "betaC:1:a" "raa-cor:3" by blast
3587
3588lemmas "β→C" = "betaC:1:a" "betaC:1:b"
3589
3590AOT_theorem "betaC:2:a": μ1...μn φ{μ1...μn}], κ1...κn, φ{κ1...κn}  μ1...μn φ{μ1...μn}]κ1...κn
3591proof -
3592  AOT_modally_strict {
3593    AOT_assume 1: μ1...μn φ{μ1...μn}] and 2: κ1...κn and 3: φ{κ1...κn}
3594    AOT_hence μ1...μn φ{μ1...μn}]κ1...κn
3595      using "beta-C-cor:2"[THEN "→E", OF 1, THEN "∀E"(1), THEN "≡E"(2)] by blast
3596  }
3597  AOT_thus μ1...μn φ{μ1...μn}], κ1...κn, φ{κ1...κn}  μ1...μn φ{μ1...μn}]κ1...κn
3598    by blast
3599qed
3600
3601AOT_theorem "betaC:2:b": μ1...μn φ{μ1...μn}], κ1...κn, ¬μ1...μn φ{μ1...μn}]κ1...κn  ¬φ{κ1...κn}
3602  using "betaC:2:a" "raa-cor:3" by blast
3603
3604lemmas "β←C" = "betaC:2:a" "betaC:2:b"
3605
3606AOT_theorem "eta-conversion-lemma1:1": Π  x1...xn [Π]x1...xn] = Π
3607  using "lambda-predicates:3"[axiom_inst] "∀I" "∀E"(1) "→I" by fast
3608
3609AOT_theorem "eta-conversion-lemma1:2": Π  ν1...νn [Π]ν1...νn] = Π
3610  using "eta-conversion-lemma1:1". (* TODO: spurious in the embedding *)
3611
3612(* match (τ) in "λa . ?b" ⇒ ‹match (τ') in "λa . ?b" ⇒ ‹fail›› ¦ _ ⇒ ‹ *)
3613
3614text‹Note: not explicitly part of PLM.›
3615AOT_theorem id_sym: assumes τ = τ' shows τ' = τ
3616  using "rule=E"[where φ="λ τ' . «τ' = τ»", rotated 1, OF assms]
3617        "=I"(1)[OF "t=t-proper:1"[THEN "→E", OF assms]] by auto
3618declare id_sym[sym]
3619
3620text‹Note: not explicitly part of PLM.›
3621AOT_theorem id_trans: assumes τ = τ' and τ' = τ'' shows τ = τ''
3622  using "rule=E" assms by blast
3623declare id_trans[trans]
3624
3625method "ηC" for Π :: <'a::{AOT_Term_id_2,AOT_κs}> = (match conclusion in "[v  τ{Π} = τ'{Π}]" for v τ τ'  3626rule "rule=E"[rotated 1, OF "eta-conversion-lemma1:2"[THEN "→E", of v "«[Π]»", symmetric]]
3627)
3628(*
3629AOT_theorem ‹[λy [λz [P]z]y → [λu [S]u]y] = [λy [P]y → [S]y]›
3630  apply ("ηC" "«[P]»") defer
3631   apply ("ηC" "«[S]»") defer
3632  oops
3633*)
3634(* TODO: proper representation of eta_conversion_lemma2 *)
3635
3636AOT_theorem "sub-des-lam:1": z1...zn  χ{z1...zn, ιx φ{x}}] & ιx φ{x} = ιx ψ{x}  z1...zn χ{z1...zn, ιx φ{x}}] = z1...zn χ{z1...zn, ιx ψ{x}}]
3637proof(rule "→I")
3638  AOT_assume A: z1...zn  χ{z1...zn, ιx φ{x}}] & ιx φ{x} = ιx ψ{x}
3639  AOT_show z1...zn χ{z1...zn, ιx φ{x}}] = z1...zn χ{z1...zn, ιx ψ{x}}]
3640    using "rule=E"[where φ="λ τ . «z1...zn χ{z1...zn, ιx φ{x}}] = z1...zn χ{z1...zn, τ}]»",
3641               OF "=I"(1)[OF A[THEN "&E"(1)]], OF A[THEN "&E"(2)]]
3642    by blast
3643qed
3644
3645AOT_theorem "sub-des-lam:2": ιx φ{x} = ιx ψ{x}  χ{ιx φ{x}} = χ{ιx ψ{x}} for χ :: ‹κ  𝗈›
3646  using "rule=E"[where φ="λ τ . «χ{ιx φ{x}} = χ{τ}»", OF "=I"(1)[OF "log-prop-prop:2"]] "→I" by blast
3647
3648AOT_theorem "prop-equiv": F = G  x (x[F]  x[G])
3649proof(rule "≡I"; rule "→I")
3650  AOT_assume F = G
3651  AOT_thus x (x[F]  x[G])
3652    by (rule "rule=E"[rotated]) (fact "oth-class-taut:3:a"[THEN GEN])
3653next
3654  AOT_assume x (x[F]  x[G])
3655  AOT_hence x[F]  x[G] for x using "∀E" by blast
3656  AOT_hence (x[F]  x[G]) for x using "en-eq:6[1]"[THEN "≡E"(1)] by blast
3657  AOT_hence x (x[F]  x[G]) by (rule GEN)
3658  AOT_hence x (x[F]  x[G]) using BF[THEN "→E"] by fast
3659  AOT_thus "F = G" using "p-identity-thm2:1"[THEN "≡E"(2)] by blast
3660qed
3661
3662AOT_theorem "relations:1":
3663  assumes INSTANCE_OF_CQT_2(φ)
3664  shows F x1...∀xn ([F]x1...xn  φ{x1...xn})
3665  apply (rule "∃I"(1)[where τ="«x1...xn φ{x1...xn}]»"])
3666  using "cqt:2[lambda]"[OF assms, axiom_inst] "beta-C-cor:2"[THEN "→E", THEN RN] by blast+
3667
3668AOT_theorem "relations:2":
3669  assumes INSTANCE_OF_CQT_2(φ)
3670  shows F x ([F]x  φ{x})
3671  using "relations:1" assms by blast
3672
3673AOT_theorem "block-paradox:1": ¬x G (x[G] & ¬[G]x)]
3674proof(rule RAA(2))
3675  let ="λ τ. «G (τ[G] & ¬[G]τ)»"
3676  AOT_assume A: x « x»]
3677  AOT_have x (A!x & F (x[F]  F = x « x»]))
3678    using "A-objects"[axiom_inst] by fast
3679  then AOT_obtain a where ξ: A!a & F (a[F]  F = x « x»])
3680    using "∃E"[rotated] by blast
3681  AOT_show ¬x G (x[G] & ¬[G]x)]
3682  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3683    AOT_assume B: x « x»]a
3684    AOT_hence G (a[G] & ¬[G]a) using "β→C" A by blast
3685    then AOT_obtain P where a[P] & ¬[P]a using "∃E"[rotated] by blast
3686    moreover AOT_have P = x « x»]
3687      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3688    ultimately AOT_have ¬x « x»]a
3689      using "rule=E" "&E"(2) by fast
3690    AOT_thus ¬x G (x[G] & ¬[G]x)] using B RAA by blast
3691  next
3692    AOT_assume B: ¬x « x»]a
3693    AOT_hence ¬G (a[G] & ¬[G]a) using "β←C" "cqt:2[const_var]"[of a, axiom_inst] A by blast
3694    AOT_hence C: G ¬(a[G] & ¬[G]a) using "cqt-further:4"[THEN "→E"] by blast
3695    AOT_have G (a[G]  [G]a)
3696      by (AOT_subst_old "λ Π . «a[Π]  [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3697         (auto simp: "oth-class-taut:1:a" C)
3698    AOT_hence ax « x»]  x « x»]a using "∀E" A by blast
3699    moreover AOT_have ax « x»] using ξ[THEN "&E"(2), THEN "∀E"(1), OF A, THEN "≡E"(2)]
3700      using "=I"(1)[OF A] by blast
3701    ultimately AOT_show ¬x G (x[G] & ¬[G]x)] using B "→E" RAA by blast
3702  qed
3703qed(simp)
3704
3705AOT_theorem "block-paradox:2": ¬F x([F]x  G(x[G] & ¬[G]x))
3706proof(rule RAA(2))
3707  AOT_assume F x ([F]x  G (x[G] & ¬[G]x))
3708  then AOT_obtain F where F_prop: x ([F]x  G (x[G] & ¬[G]x)) using "∃E"[rotated] by blast
3709  AOT_have x (A!x & G (x[G]  G = F))
3710    using "A-objects"[axiom_inst] by fast
3711  then AOT_obtain a where ξ: A!a & G (a[G]  G = F)
3712    using "∃E"[rotated] by blast
3713  AOT_show ¬F x([F]x  G(x[G] & ¬[G]x))
3714  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3715    AOT_assume B: [F]a
3716    AOT_hence G (a[G] & ¬[G]a) using F_prop[THEN "∀E"(2), THEN "≡E"(1)] by blast
3717    then AOT_obtain P where a[P] & ¬[P]a using "∃E"[rotated] by blast
3718    moreover AOT_have P = F
3719      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3720    ultimately AOT_have ¬[F]a
3721      using "rule=E" "&E"(2) by fast
3722    AOT_thus ¬F x([F]x  G(x[G] & ¬[G]x)) using B RAA by blast
3723  next
3724    AOT_assume B: ¬[F]a
3725    AOT_hence ¬G (a[G] & ¬[G]a)
3726      using "oth-class-taut:4:b"[THEN "≡E"(1), OF F_prop[THEN "∀E"(2)[of _ _ a]], THEN "≡E"(1)] by simp
3727    AOT_hence C: G ¬(a[G] & ¬[G]a) using "cqt-further:4"[THEN "→E"] by blast
3728    AOT_have G (a[G]  [G]a)
3729      by (AOT_subst_old "λ Π . «a[Π]  [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3730         (auto simp: "oth-class-taut:1:a" C)
3731    AOT_hence a[F]  [F]a using "∀E" by blast
3732    moreover AOT_have a[F] using ξ[THEN "&E"(2), THEN "∀E"(2), of F, THEN "≡E"(2)]
3733      using "=I"(2) by blast
3734    ultimately AOT_show ¬F x([F]x  G(x[G] & ¬[G]x)) using B "→E" RAA by blast
3735  qed
3736qed(simp)
3737
3738AOT_theorem "block-paradox:3": ¬y z z = y]
3739proof(rule RAA(2))
3740  AOT_assume θ: y z z = y]
3741  AOT_have x (A!x & F (x[F]  y(F = z z = y] & ¬y[F])))
3742    using "A-objects"[axiom_inst] by force
3743  then AOT_obtain a where a_prop: A!a & F (a[F]  y (F = z z = y] & ¬y[F]))
3744    using "∃E"[rotated] by blast
3745  AOT_have ζ: az z = a]  y (z z = a] = z z = y] & ¬yz z = a])
3746    using θ[THEN "∀E"(2)] a_prop[THEN "&E"(2), THEN "∀E"(1)] by blast
3747  AOT_show ¬y z z = y]
3748  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3749    AOT_assume A: az z = a]
3750    AOT_hence y (z z = a] = z z = y] & ¬yz z = a])
3751      using ζ[THEN "≡E"(1)] by blast
3752    then AOT_obtain b where b_prop: z z = a] = z z = b] & ¬bz z = a]
3753      using "∃E"[rotated] by blast
3754    moreover AOT_have a = a by (rule "=I")
3755    moreover AOT_have z z = a] using θ "∀E" by blast
3756    moreover AOT_have a using "cqt:2[const_var]"[axiom_inst] .
3757    ultimately AOT_have z z = a]a using "β←C" by blast
3758    AOT_hence z z = b]a using "rule=E" b_prop[THEN "&E"(1)] by fast
3759    AOT_hence a = b using "β→C" by blast
3760    AOT_hence bz z = a] using A "rule=E" by fast
3761    AOT_thus ¬y z z = y] using b_prop[THEN "&E"(2)] RAA by blast
3762  next
3763    AOT_assume A: ¬az z = a]
3764    AOT_hence ¬y (z z = a] = z z = y] & ¬yz z = a])
3765      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3766    AOT_hence y ¬(z z = a] = z z = y] & ¬yz z = a])
3767      using "cqt-further:4"[THEN "→E"] by blast
3768    AOT_hence ¬(z z = a] = z z = a] & ¬az z = a])
3769      using "∀E" by blast
3770    AOT_hence z z = a] = z z = a]  az z = a]
3771      by (metis "&I" "deduction-theorem" "raa-cor:4")
3772    AOT_hence az z = a] using "=I"(1) θ[THEN "∀E"(2)] "→E" by blast
3773    AOT_thus ¬y z z = y] using A RAA by blast
3774  qed
3775qed(simp)
3776
3777AOT_theorem "block-paradox:4": ¬y F x([F]x  x = y)
3778proof(rule RAA(2))
3779  AOT_assume θ: y F x([F]x  x = y)
3780  AOT_have x (A!x & F (x[F]  z (y([F]y  y = z) & ¬z[F])))
3781    using "A-objects"[axiom_inst] by force
3782  then AOT_obtain a where a_prop: A!a & F (a[F]  z (y([F]y  y = z) & ¬z[F]))
3783    using "∃E"[rotated] by blast
3784  AOT_obtain F where F_prop: x ([F]x  x = a) using θ[THEN "∀E"(2)] "∃E"[rotated] by blast
3785  AOT_have ζ: a[F]  z (y ([F]y  y = z) & ¬z[F])
3786    using a_prop[THEN "&E"(2), THEN "∀E"(2)] by blast
3787  AOT_show ¬y F x([F]x  x = y)
3788  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3789    AOT_assume A: a[F]
3790    AOT_hence z (y ([F]y  y = z) & ¬z[F])
3791      using ζ[THEN "≡E"(1)] by blast
3792    then AOT_obtain b where b_prop: y ([F]y  y = b) & ¬b[F]
3793      using "∃E"[rotated] by blast
3794    moreover AOT_have [F]a using F_prop[THEN "∀E"(2), THEN "≡E"(2)] "=I"(2) by blast
3795    ultimately AOT_have a = b using "∀E"(2) "≡E"(1) "&E" by fast
3796    AOT_hence a = b using "β→C" by blast
3797    AOT_hence b[F] using A "rule=E" by fast
3798    AOT_thus ¬y F x([F]x  x = y) using b_prop[THEN "&E"(2)] RAA by blast
3799  next
3800    AOT_assume A: ¬a[F]
3801    AOT_hence ¬z (y ([F]y  y = z) & ¬z[F])
3802      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3803    AOT_hence z ¬(y ([F]y  y = z) & ¬z[F])
3804      using "cqt-further:4"[THEN "→E"] by blast
3805    AOT_hence ¬(y ([F]y  y = a) & ¬a[F])
3806      using "∀E" by blast
3807    AOT_hence y ([F]y  y = a)  a[F]
3808      by (metis "&I" "deduction-theorem" "raa-cor:4")
3809    AOT_hence a[F] using F_prop "→E" by blast
3810    AOT_thus ¬y F x([F]x  x = y) using A RAA by blast
3811  qed
3812qed(simp)
3813
3814AOT_theorem "block-paradox:5": ¬Fxy([F]xy  y = x)
3815proof(rule "raa-cor:2")
3816  AOT_assume Fxy([F]xy  y = x)
3817  then AOT_obtain F where F_prop: xy([F]xy  y = x) using "∃E"[rotated] by blast
3818  {
3819    fix x
3820    AOT_have 1: y([F]xy  y = x) using F_prop "∀E" by blast
3821    AOT_have 2: z [F]xz] by "cqt:2[lambda]"
3822    moreover AOT_have y(z [F]xz]y  y = x)
3823    proof(rule "∀I")
3824      fix y
3825      AOT_have z [F]xz]y  [F]xy
3826        using "beta-C-meta"[THEN "→E"] 2 by fast
3827      also AOT_have ...  y = x using 1 "∀E"
3828        by fast
3829      finally AOT_show z [F]xz]y  y = x.
3830    qed
3831    ultimately AOT_have Fy([F]y  y = x)
3832      using "∃I" by fast
3833  }
3834  AOT_hence xFy([F]y  y = x)
3835    by (rule GEN)
3836  AOT_thus xFy([F]y  y = x) & ¬xFy([F]y  y = x)
3837    using "&I" "block-paradox:4" by blast
3838qed
3839
3840AOT_act_theorem "block-paradox2:1": x [G]x  ¬x [G]ιy (y = x & H (x[H] & ¬[H]x))]
3841proof(rule "→I"; rule "raa-cor:2")
3842  AOT_assume antecedant: x [G]x
3843  AOT_have Lemma: x ([G]ιy(y = x & H (x[H] & ¬[H]x))  H (x[H] & ¬[H]x))
3844  proof(rule GEN)
3845    fix x
3846    AOT_have A: [G]ιy (y = x & H (x[H] & ¬[H]x))  ∃!y (y = x & H (x[H] & ¬[H]x))
3847    proof(rule "≡I"; rule "→I")
3848      AOT_assume [G]ιy (y = x & H (x[H] & ¬[H]x))
3849      AOT_hence ιy (y = x & H (x[H] & ¬[H]x))
3850        using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
3851      AOT_thus ∃!y (y = x & H (x[H] & ¬[H]x))
3852        using "1-exists:1"[THEN "≡E"(1)] by blast
3853    next
3854      AOT_assume A: ∃!y (y = x & H (x[H] & ¬[H]x))
3855      AOT_obtain a where a_1: a = x & H (x[H] & ¬[H]x) and a_2: z (z = x & H (x[H] & ¬[H]x)  z = a)
3856        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3857      AOT_have a_3: [G]a
3858        using antecedant "∀E" by blast
3859      AOT_show [G]ιy (y = x & H (x[H] & ¬[H]x))
3860        apply (rule "russell-axiom[exe,1].russell-axiom"[THEN "≡E"(2)])
3861        apply (rule "∃I"(2))
3862        using a_1 a_2 a_3 "&I" by blast
3863    qed
3864    also AOT_have B: ...  H (x[H] & ¬[H]x)
3865    proof (rule "≡I"; rule "→I")
3866      AOT_assume A: ∃!y (y = x & H (x[H] & ¬[H]x))
3867      AOT_obtain a where a = x & H (x[H] & ¬[H]x)
3868        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3869      AOT_thus H (x[H] & ¬[H]x) using "&E" by blast
3870    next
3871      AOT_assume H (x[H] & ¬[H]x)
3872      AOT_hence x = x & H (x[H] & ¬[H]x)
3873        using "id-eq:1" "&I" by blast
3874      moreover AOT_have z (z = x & H (x[H] & ¬[H]x)  z = x)
3875        by (simp add: "Conjunction Simplification"(1) "universal-cor")
3876      ultimately AOT_show ∃!y (y = x & H (x[H] & ¬[H]x))
3877        using "uniqueness:1"[THEN "≡dfI"] "&I" "∃I"(2) by fast
3878    qed
3879    finally AOT_show ([G]ιy(y = x & H (x[H] & ¬[H]x))  H (x[H] & ¬[H]x)) .
3880  qed
3881
3882  AOT_assume A: x [G]ιy (y = x & H (x[H] & ¬[H]x))]
3883  AOT_have θ: x (x [G]ιy (y = x & H (x[H] & ¬[H]x))]x  [G]ιy(y = x & H (x[H] & ¬[H]x)))
3884    using "beta-C-meta"[THEN "→E", OF A] "∀I" by fast
3885  AOT_have x (x [G]ιy (y = x & H (x[H] & ¬[H]x))]x  H (x[H] & ¬[H]x))
3886    using θ Lemma "cqt-basic:10"[THEN "→E"] "&I" by fast
3887  AOT_hence F x ([F]x  H (x[H] & ¬[H]x))
3888    using "∃I"(1) A by fast
3889  AOT_thus (F x ([F]x  H (x[H] & ¬[H]x))) & (¬F x ([F]x  H (x[H] & ¬[H]x)))
3890    using "block-paradox:2" "&I" by blast
3891qed
3892
3893AOT_act_theorem "block-paradox2:2": G ¬x [G]ιy (y = x & H (x[H] & ¬[H]x))]
3894proof(rule "∃I"(1))
3895  AOT_have 0: x p (p p)]
3896    by "cqt:2[lambda]"
3897  moreover AOT_have x x p (p p)]x
3898    apply (rule GEN)
3899    apply (rule "beta-C-cor:2"[THEN "→E", OF 0, THEN "∀E"(2), THEN "≡E"(2)])
3900    using "if-p-then-p" GEN by fast
3901  moreover AOT_have G (x [G]x  ¬x [G]ιy (y = x & H (x[H] & ¬[H]x))])
3902      using "block-paradox2:1" "∀I" by fast
3903  ultimately AOT_show ¬x x p (p p)]ιy (y = x & H (x[H] & ¬[H]x))]
3904    using "∀E"(1) "→E" by blast
3905qed("cqt:2[lambda]")
3906
3907AOT_theorem propositions: p (p  φ)
3908proof(rule "∃I"(1))
3909  AOT_show (φ  φ)
3910    by (simp add: RN "oth-class-taut:3:a")
3911next
3912  AOT_show φ
3913    by (simp add: "log-prop-prop:2")
3914qed
3915
3916AOT_theorem "pos-not-equiv-ne:1": (¬x1...∀xn ([F]x1...xn  [G]x1...xn))  F  G
3917proof (rule "→I")
3918  AOT_assume ¬x1...∀xn ([F]x1...xn  [G]x1...xn)
3919  AOT_hence ¬x1...∀xn ([F]x1...xn  [G]x1...xn)
3920    using "KBasic:11"[THEN "≡E"(2)] by blast
3921  AOT_hence ¬(F = G)
3922    using "id-rel-nec-equiv:1" "modus-tollens:1" by blast
3923  AOT_thus F  G
3924    using "=-infix"[THEN "≡dfI"] by blast
3925qed
3926
3927AOT_theorem "pos-not-equiv-ne:2": (¬(φ{F}  φ{G}))  F  G
3928proof (rule "→I")
3929  AOT_modally_strict {
3930    AOT_have ¬(φ{F}  φ{G})  ¬(F = G)
3931    proof (rule "→I"; rule "raa-cor:2")
3932      AOT_assume 1: F = G
3933      AOT_hence φ{F}  φ{G} using "l-identity"[axiom_inst, THEN "→E"] by blast
3934      moreover {
3935        AOT_have G = F using 1 id_sym by blast
3936        AOT_hence φ{G}  φ{F} using "l-identity"[axiom_inst, THEN "→E"] by blast
3937      }
3938      ultimately AOT_have φ{F}  φ{G} using "≡I" by blast
3939      moreover AOT_assume ¬(φ{F}  φ{G})
3940      ultimately AOT_show (φ{F}  φ{G}) & ¬(φ{F}  φ{G})
3941        using "&I" by blast
3942    qed
3943  }
3944  AOT_hence ¬(φ{F}  φ{G})  ¬(F = G)
3945    using "RM:2[prem]" by blast
3946  moreover AOT_assume ¬(φ{F}  φ{G})
3947  ultimately AOT_have 0: ¬(F = G) using "→E" by blast
3948  AOT_have (F  G)
3949    by (AOT_subst F  G ¬(F = G))
3950       (auto simp: "=-infix" "≡Df" 0)
3951  AOT_thus F  G
3952    using "id-nec2:3"[THEN "→E"] by blast
3953qed
3954
3955AOT_theorem "pos-not-equiv-ne:2[zero]": (¬(φ{p}  φ{q}))  p  q
3956proof (rule "→I")
3957  AOT_modally_strict {
3958    AOT_have ¬(φ{p}  φ{q})  ¬(p = q)
3959    proof (rule "→I"; rule "raa-cor:2")
3960      AOT_assume 1: p = q
3961      AOT_hence φ{p}  φ{q} using "l-identity"[axiom_inst, THEN "→E"] by blast
3962      moreover {
3963        AOT_have q = p using 1 id_sym by blast
3964        AOT_hence φ{q}  φ{p} using "l-identity"[axiom_inst, THEN "→E"] by blast
3965      }
3966      ultimately AOT_have φ{p}  φ{q} using "≡I" by blast
3967      moreover AOT_assume ¬(φ{p}  φ{q})
3968      ultimately AOT_show (φ{p}  φ{q}) & ¬(φ{p}  φ{q})
3969        using "&I" by blast
3970    qed
3971  }
3972  AOT_hence ¬(φ{p}  φ{q})  ¬(p = q)
3973    using "RM:2[prem]" by blast
3974  moreover AOT_assume ¬(φ{p}  φ{q})
3975  ultimately AOT_have 0: ¬(p = q) using "→E" by blast
3976  AOT_have (p  q)
3977    by (AOT_subst p  q ¬(p = q))
3978       (auto simp: 0 "=-infix" "≡Df")
3979  AOT_thus p  q
3980    using "id-nec2:3"[THEN "→E"] by blast
3981qed
3982
3983AOT_theorem "pos-not-equiv-ne:3": (¬x1...∀xn ([F]x1...xn  [G]x1...xn))  F  G
3984  using "→I" "pos-not-equiv-ne:1"[THEN "→E"] "T◇"[THEN "→E"] by blast
3985
3986AOT_theorem "pos-not-equiv-ne:4": (¬(φ{F}  φ{G}))  F  G
3987  using "→I" "pos-not-equiv-ne:2"[THEN "→E"] "T◇"[THEN "→E"] by blast
3988
3989AOT_theorem "pos-not-equiv-ne:4[zero]": (¬(φ{p}  φ{q}))  p  q
3990  using "→I" "pos-not-equiv-ne:2[zero]"[THEN "→E"] "T◇"[THEN "→E"] by blast
3991
3992AOT_define relation_negation ::  Π" ("_-")
3993  "df-relation-negation": "[F]- =df x1...xn ¬[F]x1...xn]"
3994
3995nonterminal φneg
3996syntax "" :: "φneg  τ" ("_")
3997syntax "" :: "φneg  φ" ("'(_')")
3998
3999AOT_define relation_negation_0 :: ‹φ  φneg› ("'(_')-")
4000  "df-relation-negation[zero]": "(p)- =df  ¬p]"
4001
4002AOT_theorem "rel-neg-T:1": x1...xn ¬[Π]x1...xn]
4003  by "cqt:2[lambda]"
4004
4005AOT_theorem "rel-neg-T:1[zero]":  ¬φ]
4006  using "cqt:2[lambda0]"[axiom_inst] by blast
4007
4008AOT_theorem "rel-neg-T:2": [Π]- = x1...xn ¬[Π]x1...xn]
4009  using "=I"(1)[OF "rel-neg-T:1"]
4010  by (rule "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"])
4011
4012AOT_theorem "rel-neg-T:2[zero]": (φ)- =  ¬φ]
4013  using "=I"(1)[OF "rel-neg-T:1[zero]"]
4014  by (rule "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"])
4015
4016AOT_theorem "rel-neg-T:3": [Π]-
4017  using "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"] "rel-neg-T:1" by blast
4018
4019AOT_theorem "rel-neg-T:3[zero]": (φ)-
4020  using "log-prop-prop:2" by blast
4021(*  using "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"] "rel-neg-T:1[zero]" by blast *)
4022
4023(* Note: PLM states the zero place case twice *)
4024AOT_theorem "thm-relation-negation:1": [F]-x1...xn  ¬[F]x1...xn
4025proof -
4026  AOT_have [F]-x1...xn  x1...xn ¬[F]x1...xn]x1...xn
4027    using "rule=E"[rotated, OF "rel-neg-T:2"] "rule=E"[rotated, OF "rel-neg-T:2"[THEN id_sym]]
4028    "→I" "≡I" by fast
4029  also AOT_have ...  ¬[F]x1...xn
4030    using "beta-C-meta"[THEN "→E", OF "rel-neg-T:1"] by fast
4031  finally show ?thesis.
4032qed
4033
4034AOT_theorem "thm-relation-negation:2": ¬[F]-x1...xn  [F]x1...xn
4035  apply (AOT_subst [F]x1...xn ¬¬[F]x1...xn)
4036   apply (simp add: "oth-class-taut:3:b")
4037  apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4038  using "thm-relation-negation:1".
4039
4040AOT_theorem "thm-relation-negation:3": ((p)-)  ¬p
4041proof -
4042  AOT_have (p)- =  ¬p] using "rel-neg-T:2[zero]" by blast
4043  AOT_hence ((p)-)   ¬p]
4044    using "df-relation-negation[zero]" "log-prop-prop:2" "oth-class-taut:3:a" "rule-id-df:2:a" by blast
4045  also AOT_have  ¬p]  ¬p
4046    by (simp add: "propositions-lemma:2")
4047  finally show ?thesis.
4048qed
4049
4050AOT_theorem "thm-relation-negation:4": (¬((p)-))  p
4051  using "thm-relation-negation:3"[THEN "≡E"(1)]
4052        "thm-relation-negation:3"[THEN "≡E"(2)]
4053        "≡I" "→I" RAA by metis
4054
4055AOT_theorem "thm-relation-negation:5": [F]  [F]-
4056proof -
4057  AOT_have ¬([F] = [F]-)
4058  proof (rule RAA(2))
4059    AOT_show [F]x1...xn  [F]x1...xn for x1xn
4060      using "if-p-then-p".
4061  next
4062    AOT_assume [F] = [F]-
4063    AOT_hence [F]- = [F] using id_sym by blast
4064    AOT_hence [F]x1...xn  ¬[F]x1...xn for x1xn
4065      using "rule=E" "thm-relation-negation:1" by fast
4066    AOT_thus ¬([F]x1...xn  [F]x1...xn) for x1xn
4067      using "≡E" RAA by metis
4068  qed
4069  thus ?thesis
4070    using "≡dfI" "=-infix" by blast
4071qed
4072
4073AOT_theorem "thm-relation-negation:6": p  (p)-
4074proof -
4075  AOT_have ¬(p = (p)-)
4076  proof (rule RAA(2))
4077    AOT_show p  p
4078      using "if-p-then-p".
4079  next
4080    AOT_assume p = (p)-
4081    AOT_hence (p)- = p using id_sym by blast
4082    AOT_hence p  ¬p
4083      using "rule=E" "thm-relation-negation:3" by fast
4084    AOT_thus ¬(p  p)
4085      using "≡E" RAA by metis
4086  qed
4087  thus ?thesis
4088    using "≡dfI" "=-infix" by blast
4089qed
4090
4091AOT_theorem "thm-relation-negation:7": (p)- = (¬p)
4092  apply (rule "df-relation-negation[zero]"[THEN "=dfE"(1)])
4093  using "cqt:2[lambda0]"[axiom_inst] "rel-neg-T:2[zero]" "propositions-lemma:1" id_trans by blast+
4094
4095AOT_theorem "thm-relation-negation:8": p = q  (¬p) = (¬q)
4096proof(rule "→I")
4097  AOT_assume p = q
4098  moreover AOT_have (¬p) using "log-prop-prop:2".
4099  moreover AOT_have (¬p) = (¬p) using calculation(2) "=I" by blast
4100  ultimately AOT_show (¬p) = (¬q)
4101    using "rule=E" by fast
4102qed
4103
4104AOT_theorem "thm-relation-negation:9": p = q  (p)- = (q)-
4105proof(rule "→I")
4106  AOT_assume p = q
4107  AOT_hence (¬p) = (¬q) using "thm-relation-negation:8" "→E" by blast
4108  AOT_thus (p)- = (q)-
4109    using "thm-relation-negation:7" id_sym id_trans by metis
4110qed
4111
4112AOT_define Necessary :: ‹Π  φ› ("Necessary'(_')")
4113  "contingent-properties:1": Necessary([F]) df x1...∀xn [F]x1...xn
4114
4115AOT_define Necessary0 :: ‹φ  φ› ("Necessary0'(_')")
4116  "contingent-properties:1[zero]": Necessary0(p) df p
4117
4118AOT_define Impossible :: ‹Π  φ› ("Impossible'(_')")
4119  "contingent-properties:2": Impossible([F]) df F & x1...∀xn ¬[F]x1...xn
4120
4121AOT_define Impossible0 :: ‹φ  φ› ("Impossible0'(_')")
4122  "contingent-properties:2[zero]": Impossible0(p) df ¬p
4123
4124AOT_define NonContingent :: ‹Π  φ› ("NonContingent'(_')")
4125  "contingent-properties:3": NonContingent([F]) df Necessary([F])  Impossible([F])
4126
4127AOT_define NonContingent0 :: ‹φ  φ› ("NonContingent0'(_')")
4128  "contingent-properties:3[zero]": NonContingent0(p) df Necessary0(p)  Impossible0(p)
4129
4130AOT_define Contingent :: ‹Π  φ› ("Contingent'(_')")
4131  "contingent-properties:4": Contingent([F]) df F & ¬(Necessary([F])  Impossible([F]))
4132
4133AOT_define Contingent0 :: ‹φ  φ› ("Contingent0'(_')")
4134  "contingent-properties:4[zero]": Contingent0(p) df ¬(Necessary0(p)  Impossible0(p))
4135
4136
4137AOT_theorem "thm-cont-prop:1": NonContingent([F])  NonContingent([F]-)
4138proof (rule "≡I"; rule "→I")
4139  AOT_assume NonContingent([F])
4140  AOT_hence Necessary([F])  Impossible([F])
4141    using "≡dfE"[OF "contingent-properties:3"] by blast
4142  moreover {
4143    AOT_assume Necessary([F])
4144    AOT_hence (x1...∀xn [F]x1...xn)
4145      using "≡dfE"[OF "contingent-properties:1"] by blast
4146    moreover AOT_modally_strict {
4147      AOT_assume x1...∀xn [F]x1...xn
4148      AOT_hence [F]x1...xn for x1xn using "∀E" by blast
4149      AOT_hence ¬[F]-x1...xn for x1xn
4150        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:2" "≡E"(1))
4151      AOT_hence x1...∀xn ¬[F]-x1...xn using "∀I" by fast
4152    }
4153    ultimately AOT_have (x1...∀xn ¬[F]-x1...xn)
4154      using "RN[prem]"[where Γ="{«x1...∀xn [F]x1...xn»}", simplified] by blast
4155    AOT_hence Impossible([F]-)
4156      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(2)]
4157      by blast
4158  }
4159  moreover {
4160    AOT_assume Impossible([F])
4161    AOT_hence (x1...∀xn ¬[F]x1...xn)
4162      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)]
4163      by blast
4164    moreover AOT_modally_strict {
4165      AOT_assume x1...∀xn ¬[F]x1...xn
4166      AOT_hence ¬[F]x1...xn for x1xn using "∀E" by blast
4167      AOT_hence [F]-x1...xn for x1xn
4168        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(1))
4169      AOT_hence x1...∀xn [F]-x1...xn using "∀I" by fast
4170    }
4171    ultimately AOT_have (x1...∀xn [F]-x1...xn)
4172      using "RN[prem]"[where Γ="{«x1...∀xn ¬[F]x1...xn»}"] by blast
4173    AOT_hence Necessary([F]-)
4174      using "≡dfI"[OF "contingent-properties:1"] by blast
4175  }
4176  ultimately AOT_have Necessary([F]-)  Impossible([F]-)
4177    using "∨E"(1) "∨I" "→I" by metis
4178  AOT_thus NonContingent([F]-)
4179    using "≡dfI"[OF "contingent-properties:3"] by blast
4180next
4181  AOT_assume NonContingent([F]-)
4182  AOT_hence Necessary([F]-)  Impossible([F]-)
4183    using "≡dfE"[OF "contingent-properties:3"] by blast
4184  moreover {
4185    AOT_assume Necessary([F]-)
4186    AOT_hence (x1...∀xn [F]-x1...xn)
4187      using "≡dfE"[OF "contingent-properties:1"] by blast
4188    moreover AOT_modally_strict {
4189      AOT_assume x1...∀xn [F]-x1...xn
4190      AOT_hence [F]-x1...xn for x1xn using "∀E" by blast
4191      AOT_hence ¬[F]x1...xn for x1xn
4192        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(2))
4193      AOT_hence x1...∀xn ¬[F]x1...xn using "∀I" by fast
4194    }
4195    ultimately AOT_have x1...∀xn ¬[F]x1...xn
4196      using "RN[prem]"[where Γ="{«x1...∀xn [F]-x1...xn»}"] by blast
4197    AOT_hence Impossible([F])
4198      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(2)]
4199      by blast
4200  }
4201  moreover {
4202    AOT_assume Impossible([F]-)
4203    AOT_hence (x1...∀xn ¬[F]-x1...xn)
4204      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(1)]
4205      by blast
4206    moreover AOT_modally_strict {
4207      AOT_assume x1...∀xn ¬[F]-x1...xn
4208      AOT_hence ¬[F]-x1...xn for x1xn using "∀E" by blast
4209      AOT_hence [F]x1...xn for x1xn 
4210        using "thm-relation-negation:1"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4211              "useful-tautologies:1"[THEN "→E"] by blast
4212      AOT_hence x1...∀xn [F]x1...xn using "∀I" by fast
4213    }
4214    ultimately AOT_have (x1...∀xn [F]x1...xn)
4215      using "RN[prem]"[where Γ="{«x1...∀xn ¬[F]-x1...xn»}"] by blast
4216    AOT_hence Necessary([F])
4217      using "≡dfI"[OF "contingent-properties:1"] by blast
4218  }
4219  ultimately AOT_have Necessary([F])  Impossible([F])
4220    using "∨E"(1) "∨I" "→I" by metis
4221  AOT_thus NonContingent([F])
4222    using "≡dfI"[OF "contingent-properties:3"] by blast
4223qed
4224
4225AOT_theorem "thm-cont-prop:2": Contingent([F])  x [F]x & x ¬[F]x
4226proof -
4227  AOT_have Contingent([F])  ¬(Necessary([F])  Impossible([F]))
4228    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]]
4229    by blast
4230  also AOT_have ...  ¬Necessary([F]) & ¬Impossible([F])
4231    using "oth-class-taut:5:d" by fastforce
4232  also AOT_have ...  ¬Impossible([F]) & ¬Necessary([F])
4233    by (simp add: "Commutativity of &")
4234  also AOT_have ...  x [F]x & ¬Necessary([F])
4235  proof (rule "oth-class-taut:4:e"[THEN "→E"])
4236    AOT_have ¬Impossible([F])  ¬¬ x [F]x
4237      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4238      apply (AOT_subst x [F]x ¬ x ¬[F]x)
4239       apply (simp add: "conventions:4" "≡Df")
4240      apply (AOT_subst (reverse) ¬¬x ¬[F]x x ¬[F]x)
4241       apply (simp add: "oth-class-taut:3:b")
4242      using "contingent-properties:2"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]] by blast
4243    also AOT_have ...  x [F]x
4244      using "conventions:5"[THEN "≡Df", symmetric] by blast
4245    finally AOT_show ¬Impossible([F])  x [F]x .
4246  qed
4247  also AOT_have ...  x [F]x & x ¬[F]x
4248  proof (rule "oth-class-taut:4:f"[THEN "→E"])
4249    AOT_have ¬Necessary([F])  ¬¬x ¬[F]x
4250      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4251      apply (AOT_subst x ¬[F]x ¬ x ¬¬[F]x)
4252       apply (simp add: "conventions:4" "≡Df")
4253      apply (AOT_subst (reverse) ¬¬[F]x [F]x bound: x)
4254       apply (simp add: "oth-class-taut:3:b")
4255      apply (AOT_subst (reverse) ¬¬x [F]x x [F]x)
4256      by (auto simp: "oth-class-taut:3:b" "contingent-properties:1" "≡Df")
4257    also AOT_have ...  x ¬[F]x
4258      using "conventions:5"[THEN "≡Df", symmetric] by blast
4259    finally AOT_show ¬Necessary([F])  x ¬[F]x.
4260  qed
4261  finally show ?thesis.
4262qed
4263
4264AOT_theorem "thm-cont-prop:3": Contingent([F])  Contingent([F]-) for F::<κ> AOT_var›
4265proof -
4266  {
4267    fix Π :: <κ>
4268    AOT_assume Π
4269    moreover AOT_have F (Contingent([F])  x [F]x & x ¬[F]x)
4270      using "thm-cont-prop:2" GEN by fast
4271    ultimately  AOT_have Contingent([Π])  x [Π]x & x ¬[Π]x
4272      using "thm-cont-prop:2" "∀E" by fast
4273  } note 1 = this
4274  AOT_have Contingent([F])  x [F]x & x ¬[F]x
4275    using "thm-cont-prop:2" by blast
4276  also AOT_have ...  x ¬[F]x & x [F]x
4277    by (simp add: "Commutativity of &")
4278  also AOT_have ...  x [F]-x & x [F]x
4279    by (AOT_subst [F]-x ¬[F]x bound: x)
4280       (auto simp: "thm-relation-negation:1" "oth-class-taut:3:a")
4281  also AOT_have ...  x [F]-x & x ¬[F]-x
4282    by (AOT_subst (reverse) [F]x ¬[F]-x bound: x)
4283       (auto simp: "thm-relation-negation:2" "oth-class-taut:3:a")
4284  also AOT_have ...  Contingent([F]-)
4285    using 1[OF "rel-neg-T:3", symmetric] by blast
4286  finally show ?thesis.
4287qed
4288
4289AOT_define concrete_if_concrete :: ‹Π› ("L")  L_def: L =df x E!x  E!x]
4290
4291AOT_theorem "thm-noncont-e-e:1": Necessary(L)
4292proof -
4293  AOT_modally_strict {
4294    fix x
4295    AOT_have x E!x  E!x] by "cqt:2[lambda]"
4296    moreover AOT_have x using "cqt:2[const_var]"[axiom_inst] by blast
4297    moreover AOT_have E!x  E!x using "if-p-then-p" by blast
4298    ultimately AOT_have x E!x  E!x]x
4299      using "β←C" by blast
4300  }
4301  AOT_hence 0: x x E!x  E!x]x
4302    using RN GEN by blast
4303  show ?thesis
4304    apply (rule "=dfI"(2)[OF L_def])
4305     apply "cqt:2[lambda]"
4306    by (rule "contingent-properties:1"[THEN "≡dfI", OF 0])
4307qed
4308
4309AOT_theorem "thm-noncont-e-e:2": Impossible([L]-)
4310proof -
4311  AOT_modally_strict {
4312    fix x
4313
4314    AOT_have 0: F (¬[F]-x  [F]x)
4315      using "thm-relation-negation:2" GEN by fast
4316    AOT_have ¬x E!x  E!x]-x  x E!x  E!x]x
4317      by (rule 0[THEN "∀E"(1)]) "cqt:2[lambda]"
4318    moreover {
4319      AOT_have x E!x  E!x] by "cqt:2[lambda]"
4320      moreover AOT_have x using "cqt:2[const_var]"[axiom_inst] by blast
4321      moreover AOT_have E!x  E!x using "if-p-then-p" by blast
4322      ultimately AOT_have x E!x  E!x]x
4323        using "β←C" by blast
4324    }
4325    ultimately AOT_have ¬x E!x  E!x]-x
4326      using "≡E" by blast
4327  }
4328  AOT_hence 0: x ¬x E!x  E!x]-x
4329    using RN GEN by fast
4330  show ?thesis
4331    apply (rule "=dfI"(2)[OF L_def])
4332     apply "cqt:2[lambda]"
4333    apply (rule "contingent-properties:2"[THEN "≡dfI"]; rule "&I")
4334     using "rel-neg-T:3"
4335     apply blast
4336    using 0
4337    by blast
4338qed
4339
4340AOT_theorem "thm-noncont-e-e:3": NonContingent(L)
4341  using "thm-noncont-e-e:1"
4342  by (rule "contingent-properties:3"[THEN "≡dfI", OF "∨I"(1)])
4343
4344AOT_theorem "thm-noncont-e-e:4": NonContingent([L]-)
4345proof -
4346  AOT_have 0: F (NonContingent([F])  NonContingent([F]-))
4347    using "thm-cont-prop:1" "∀I" by fast
4348  moreover AOT_have 1: L
4349    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4350  AOT_show NonContingent([L]-)
4351    using "∀E"(1)[OF 0, OF 1, THEN "≡E"(1), OF "thm-noncont-e-e:3"] by blast
4352qed
4353
4354AOT_theorem "thm-noncont-e-e:5": F G (F  «G::<κ>» & NonContingent([F]) & NonContingent([G]))
4355proof (rule "∃I")+
4356  {
4357    AOT_have F [F]  [F]- using "thm-relation-negation:5" GEN by fast
4358    moreover AOT_have L
4359      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4360    ultimately AOT_have L  [L]- using "∀E" by blast
4361  }
4362  AOT_thus L  [L]- & NonContingent(L) & NonContingent([L]-)
4363    using "thm-noncont-e-e:3" "thm-noncont-e-e:4" "&I" by metis
4364next
4365  AOT_show [L]-
4366    using "rel-neg-T:3" by blast
4367next
4368  AOT_show L
4369      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4370qed
4371
4372AOT_theorem "lem-cont-e:1": x ([F]x & ¬[F]x)  x (¬[F]x & [F]x)
4373proof -
4374  AOT_have x ([F]x & ¬[F]x)  x ([F]x & ¬[F]x)
4375    using "BF◇" "CBF◇" "≡I" by blast
4376  also AOT_have   x ([F]x &  ¬[F]x)
4377    by (AOT_subst ([F]x & ¬[F]x) [F]x &  ¬[F]x bound: x)
4378       (auto simp: "S5Basic:11" "cqt-further:7")
4379  also AOT_have   x (¬[F]x & [F]x)
4380    by (AOT_subst ¬[F]x & [F]x  [F]x & ¬[F]x bound: x)
4381       (auto simp: "Commutativity of &" "cqt-further:7")
4382  also AOT_have   x (¬[F]x & [F]x)
4383    by (AOT_subst (¬[F]x & [F]x) ¬[F]x & [F]x bound: x)
4384       (auto simp: "S5Basic:11" "oth-class-taut:3:a")
4385  also AOT_have   x (¬[F]x & [F]x)
4386    using "BF◇" "CBF◇" "≡I" by fast
4387  finally show ?thesis.
4388qed
4389
4390AOT_theorem "lem-cont-e:2": x ([F]x & ¬[F]x)  x ([F]-x & ¬[F]-x)
4391proof -
4392  AOT_have x ([F]x & ¬[F]x)  x (¬[F]x & [F]x)
4393    using "lem-cont-e:1".
4394  also AOT_have   x ([F]-x & ¬[F]-x)
4395    apply (AOT_subst ¬[F]-x [F]x bound: x)
4396     apply (simp add: "thm-relation-negation:2")
4397    apply (AOT_subst [F]-x ¬[F]x bound: x)
4398     apply (simp add: "thm-relation-negation:1")
4399    by (simp add: "oth-class-taut:3:a")
4400  finally show ?thesis.
4401qed
4402
4403AOT_theorem "thm-cont-e:1": x (E!x & ¬E!x)
4404proof (rule "CBF◇"[THEN "→E"])
4405  AOT_have x (E!x & ¬𝒜E!x) using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] by blast
4406  then AOT_obtain a where (E!a & ¬𝒜E!a) using "∃E"[rotated] by blast
4407  AOT_hence θ: E!a & ¬𝒜E!a
4408    using "KBasic2:3"[THEN "→E"] by blast
4409  AOT_have ξ: E!a & 𝒜¬E!a
4410    by (AOT_subst  𝒜¬E!a ¬𝒜E!a)
4411       (auto simp: "logic-actual-nec:1"[axiom_inst] θ)
4412  AOT_have ζ: E!a & 𝒜¬E!a
4413    by (AOT_subst 𝒜¬E!a 𝒜¬E!a)
4414       (auto simp add: "Act-Sub:4" ξ)
4415  AOT_hence E!a & ¬E!a
4416    using "&E" "&I" "Act-Sub:3"[THEN "→E"] by blast
4417  AOT_hence (E!a & ¬E!a) using "S5Basic:11"[THEN "≡E"(2)] by simp
4418  AOT_thus x (E!x & ¬E!x) using "∃I"(2) by fast
4419qed
4420
4421AOT_theorem "thm-cont-e:2": x (¬E!x & E!x)
4422proof -
4423  AOT_have F (x ([F]x & ¬[F]x)  x (¬[F]x & [F]x))
4424    using "lem-cont-e:1" GEN by fast
4425  AOT_hence (x (E!x & ¬E!x)  x (¬E!x & E!x))
4426    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4427  thus ?thesis using "thm-cont-e:1" "≡E" by blast
4428qed
4429
4430AOT_theorem "thm-cont-e:3": x E!x
4431proof (rule "CBF◇"[THEN "→E"])
4432  AOT_obtain a where (E!a & ¬E!a)
4433    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4434  AOT_hence E!a
4435    using "KBasic2:3"[THEN "→E", THEN "&E"(1)] by blast
4436  AOT_thus x E!x using "∃I" by fast
4437qed
4438
4439AOT_theorem "thm-cont-e:4": x ¬E!x
4440proof (rule "CBF◇"[THEN "→E"])
4441  AOT_obtain a where (E!a & ¬E!a)
4442    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4443  AOT_hence ¬E!a
4444    using "KBasic2:3"[THEN "→E", THEN "&E"(2)] by blast
4445  AOT_hence ¬E!a
4446    using "4◇"[THEN "→E"] by blast
4447  AOT_thus x ¬E!x using "∃I" by fast
4448qed
4449
4450AOT_theorem "thm-cont-e:5": Contingent([E!])
4451proof -
4452  AOT_have F (Contingent([F])  x [F]x & x ¬[F]x)
4453    using "thm-cont-prop:2" GEN by fast
4454  AOT_hence Contingent([E!])  x E!x & x ¬E!x
4455    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4456  thus ?thesis
4457    using "thm-cont-e:3" "thm-cont-e:4" "≡E"(2) "&I" by blast
4458qed
4459
4460AOT_theorem "thm-cont-e:6": Contingent([E!]-)
4461proof -
4462  AOT_have F (Contingent([«F::<κ>»])  Contingent([F]-))
4463    using "thm-cont-prop:3" GEN by fast
4464  AOT_hence Contingent([E!])  Contingent([E!]-)
4465    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4466  thus ?thesis using "thm-cont-e:5" "≡E" by blast
4467qed
4468
4469AOT_theorem "thm-cont-e:7": FG (Contingent([«F::<κ>»]) & Contingent([G]) & F  G)
4470proof (rule "∃I")+
4471  AOT_have F [«F::<κ>»]  [F]- using "thm-relation-negation:5" GEN by fast
4472  AOT_hence [E!]  [E!]-
4473    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4474  AOT_thus Contingent([E!]) & Contingent([E!]-) & [E!]  [E!]-
4475    using "thm-cont-e:5" "thm-cont-e:6" "&I" by metis
4476next
4477  AOT_show E!-
4478    by (fact AOT)
4479next
4480  AOT_show E! by (fact "cqt:2[concrete]"[axiom_inst])
4481qed
4482
4483AOT_theorem "property-facts:1": NonContingent([F])  ¬G (Contingent([G]) & G = F)
4484proof (rule "→I"; rule "raa-cor:2")
4485  AOT_assume NonContingent([F])
4486  AOT_hence 1: Necessary([F])  Impossible([F])
4487    using "contingent-properties:3"[THEN "≡dfE"] by blast
4488  AOT_assume G (Contingent([G]) & G = F)
4489  then AOT_obtain G where Contingent([G]) & G = F using "∃E"[rotated] by blast
4490  AOT_hence Contingent([F]) using "rule=E" "&E" by blast
4491  AOT_hence ¬(Necessary([F])  Impossible([F]))
4492    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4493  AOT_thus (Necessary([F])  Impossible([F])) & ¬(Necessary([F])  Impossible([F]))
4494    using 1 "&I" by blast
4495qed
4496
4497AOT_theorem "property-facts:2": Contingent([F])  ¬G (NonContingent([G]) & G = F)
4498proof (rule "→I"; rule "raa-cor:2")
4499  AOT_assume Contingent([F])
4500  AOT_hence 1: ¬(Necessary([F])  Impossible([F]))
4501    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4502  AOT_assume G (NonContingent([G]) & G = F)
4503  then AOT_obtain G where NonContingent([G]) & G = F using "∃E"[rotated] by blast
4504  AOT_hence NonContingent([F]) using "rule=E" "&E" by blast
4505  AOT_hence Necessary([F])  Impossible([F])
4506    using "contingent-properties:3"[THEN "≡dfE"] by blast
4507  AOT_thus (Necessary([F])  Impossible([F])) & ¬(Necessary([F])  Impossible([F]))
4508    using 1 "&I" by blast
4509qed
4510
4511AOT_theorem "property-facts:3": L  [L]- & L  E! & L  E!- & [L]-  [E!]- & E!  [E!]-
4512proof -
4513  AOT_have noneqI: Π  Π' if φ{Π} and ¬φ{Π'} for φ and Π Π' :: <κ>
4514    apply (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
4515    using "rule=E"[where φ=φ and τ=Π and σ = Π'] that "&I" by blast
4516  AOT_have contingent_denotes: Π if Contingent([Π]) for Π :: <κ>
4517    using that "contingent-properties:4"[THEN "≡dfE", THEN "&E"(1)] by blast
4518  AOT_have not_noncontingent_if_contingent: ¬NonContingent([Π]) if Contingent([Π]) for Π :: <κ>
4519  proof(rule RAA(2))
4520    AOT_show ¬(Necessary([Π])  Impossible([Π]))
4521      using that "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF contingent_denotes[OF that], THEN "≡E"(1)] by blast
4522  next
4523    AOT_assume NonContingent([Π])
4524    AOT_thus Necessary([Π])  Impossible([Π])
4525      using "contingent-properties:3"[THEN "≡dfE"] by blast
4526  qed
4527
4528  show ?thesis
4529  proof (safe intro!: "&I")
4530    AOT_show L  [L]-
4531      apply (rule "=dfI"(2)[OF L_def])
4532       apply "cqt:2[lambda]"
4533      apply (rule "∀E"(1)[where φ="λ Π . «Π  [Π]-»"])
4534       apply (rule GEN) apply (fact AOT)
4535      by "cqt:2[lambda]"
4536  next
4537    AOT_show L  E!
4538      apply (rule noneqI)
4539      using "thm-noncont-e-e:3" not_noncontingent_if_contingent[OF "thm-cont-e:5"]
4540      by auto
4541  next
4542    AOT_show L  E!-
4543      apply (rule noneqI)
4544      using "thm-noncont-e-e:3" apply fast
4545      apply (rule not_noncontingent_if_contingent)
4546      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π])  Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4547      using "thm-cont-prop:3" GEN apply fast
4548      using "thm-cont-e:5" by fast+
4549  next
4550    AOT_show [L]-  E!-
4551      apply (rule noneqI)
4552      using "thm-noncont-e-e:4" apply fast
4553      apply (rule not_noncontingent_if_contingent)
4554      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π])  Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4555      using "thm-cont-prop:3" GEN apply fast
4556      using "thm-cont-e:5" by fast+
4557  next
4558    AOT_show E!  E!-
4559      apply (rule "=dfI"(2)[OF L_def])
4560       apply "cqt:2[lambda]"
4561      apply (rule "∀E"(1)[where φ="λ Π . «Π  [Π]-»"])
4562       apply (rule GEN) apply (fact AOT)
4563      by (fact "cqt:2[concrete]"[axiom_inst])
4564  qed
4565qed
4566
4567AOT_theorem "thm-cont-propos:1": NonContingent0(p)  NonContingent0(((p)-))
4568proof(rule "≡I"; rule "→I")
4569  AOT_assume NonContingent0(p)
4570  AOT_hence Necessary0(p)  Impossible0(p)
4571    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4572  moreover {
4573    AOT_assume Necessary0(p)
4574    AOT_hence 1: p using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4575    AOT_have ¬((p)-)
4576      by (AOT_subst ¬((p)-) p)
4577         (auto simp add: 1 "thm-relation-negation:4")
4578    AOT_hence Impossible0(((p)-))
4579      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4580  }
4581  moreover {
4582    AOT_assume Impossible0(p)
4583    AOT_hence 1: ¬p
4584      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4585    AOT_have ((p)-)
4586      by (AOT_subst ((p)-) ¬p) 
4587         (auto simp: 1 "thm-relation-negation:3")
4588    AOT_hence Necessary0(((p)-))
4589      by (rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4590  }
4591  ultimately AOT_have Necessary0(((p)-))  Impossible0(((p)-))
4592    using "∨E"(1) "∨I" "→I" by metis
4593  AOT_thus NonContingent0(((p)-))
4594    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4595next
4596  AOT_assume NonContingent0(((p)-))
4597  AOT_hence Necessary0(((p)-))  Impossible0(((p)-))
4598    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4599  moreover {
4600    AOT_assume Impossible0(((p)-))
4601    AOT_hence 1: ¬((p)-)
4602      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4603    AOT_have p
4604      by (AOT_subst (reverse) p ¬((p)-))
4605         (auto simp: 1 "thm-relation-negation:4")
4606    AOT_hence Necessary0(p)
4607      using "contingent-properties:1[zero]"[THEN "≡dfI"] by blast
4608  }
4609  moreover {
4610    AOT_assume Necessary0(((p)-))
4611    AOT_hence 1: ((p)-)
4612      by (rule "contingent-properties:1[zero]"[THEN "≡dfE"])
4613    AOT_have ¬p
4614      by (AOT_subst (reverse) ¬p ((p)-))
4615         (auto simp: 1 "thm-relation-negation:3")
4616    AOT_hence Impossible0(p)
4617      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4618  }
4619  ultimately AOT_have Necessary0(p)  Impossible0(p)
4620    using "∨E"(1) "∨I" "→I" by metis
4621  AOT_thus NonContingent0(p)
4622    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4623qed
4624
4625AOT_theorem "thm-cont-propos:2": Contingent0(φ)  φ & ¬φ
4626proof -
4627  AOT_have Contingent0(φ)  ¬(Necessary0(φ)  Impossible0(φ))
4628    using "contingent-properties:4[zero]"[THEN "≡Df"] by simp
4629  also AOT_have   ¬Necessary0(φ) & ¬Impossible0(φ)
4630    by (fact AOT)
4631  also AOT_have   ¬Impossible0(φ) & ¬Necessary0(φ)
4632    by (fact AOT)
4633  also AOT_have   φ & ¬φ
4634    apply (AOT_subst φ ¬¬φ)
4635     apply (simp add: "conventions:5" "≡Df")
4636    apply (AOT_subst Impossible0(φ) ¬φ)
4637     apply (simp add: "contingent-properties:2[zero]" "≡Df")
4638    apply (AOT_subst (reverse) ¬φ ¬φ)
4639     apply (simp add: "KBasic:11")
4640    apply (AOT_subst Necessary0(φ) φ)
4641     apply (simp add: "contingent-properties:1[zero]" "≡Df")
4642    by (simp add: "oth-class-taut:3:a")
4643  finally show ?thesis.
4644qed
4645
4646AOT_theorem "thm-cont-propos:3": Contingent0(p)  Contingent0(((p)-))
4647proof -
4648  AOT_have Contingent0(p)  p & ¬p using "thm-cont-propos:2".
4649  also AOT_have   ¬p & p by (fact AOT)
4650  also AOT_have   ((p)-) & p
4651    by (AOT_subst ((p)-) ¬p)
4652       (auto simp: "thm-relation-negation:3" "oth-class-taut:3:a")
4653  also AOT_have   ((p)-) & ¬((p)-)
4654    by (AOT_subst ¬((p)-) p)
4655       (auto simp: "thm-relation-negation:4" "oth-class-taut:3:a")
4656  also AOT_have   Contingent0(((p)-))
4657    using "thm-cont-propos:2"[symmetric] by blast
4658  finally show ?thesis.
4659qed
4660
4661AOT_define noncontingent_prop :: ‹φ› ("p0")
4662  p0_def: "(p0) =df (x (E!x  E!x))"
4663
4664AOT_theorem "thm-noncont-propos:1":  Necessary0((p0))
4665proof(rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4666  AOT_show (p0)
4667    apply (rule "=dfI"(2)[OF p0_def])
4668    using "log-prop-prop:2" apply simp
4669    using "if-p-then-p" RN GEN by fast
4670qed
4671
4672AOT_theorem "thm-noncont-propos:2": Impossible0(((p0)-))
4673proof(rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4674  AOT_show ¬((p0)-)
4675    apply (AOT_subst ((p0)-) ¬p0)
4676    using "thm-relation-negation:3" GEN "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4677    apply (AOT_subst (reverse) ¬¬p0 p0)
4678     apply (simp add: "oth-class-taut:3:b")
4679    apply (rule "=dfI"(2)[OF p0_def])
4680    using "log-prop-prop:2" apply simp
4681    using "if-p-then-p" RN GEN by fast
4682qed
4683
4684AOT_theorem "thm-noncont-propos:3": NonContingent0((p0))
4685  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4686  using "thm-noncont-propos:1" "∨I" by blast
4687
4688AOT_theorem "thm-noncont-propos:4": NonContingent0(((p0)-))
4689  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4690  using "thm-noncont-propos:2" "∨I" by blast
4691
4692AOT_theorem "thm-noncont-propos:5": pq (NonContingent0((p)) & NonContingent0((q)) & p  q)
4693proof(rule "∃I")+
4694  AOT_have 0: φ  (φ)- for φ
4695    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4696  AOT_thus NonContingent0((p0)) & NonContingent0(((p0)-)) & (p0)  (p0)-
4697    using "thm-noncont-propos:3" "thm-noncont-propos:4" "&I" by auto
4698qed(auto simp: "log-prop-prop:2")
4699
4700AOT_act_theorem "no-cnac": ¬x(E!x & ¬𝒜E!x)
4701proof(rule "raa-cor:2")
4702  AOT_assume x(E!x & ¬𝒜E!x)
4703  then AOT_obtain a where a: E!a & ¬𝒜E!a
4704    using "∃E"[rotated] by blast
4705  AOT_hence 𝒜¬E!a using "&E" "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
4706  AOT_hence ¬E!a using "logic-actual"[act_axiom_inst, THEN "→E"] by blast
4707  AOT_hence E!a & ¬E!a using a "&E" "&I" by blast
4708  AOT_thus p & ¬p for p using "raa-cor:1" by blast
4709qed
4710
4711AOT_theorem "pos-not-pna:1": ¬𝒜x (E!x & ¬𝒜E!x)
4712proof(rule "raa-cor:2")
4713  AOT_assume 𝒜x (E!x & ¬𝒜E!x)
4714  AOT_hence x 𝒜(E!x & ¬𝒜E!x)
4715    using "Act-Basic:10"[THEN "≡E"(1)] by blast
4716  then AOT_obtain a where 𝒜(E!a & ¬𝒜E!a) using "∃E"[rotated] by blast
4717  AOT_hence 1: 𝒜E!a & 𝒜¬𝒜E!a using "Act-Basic:2"[THEN "≡E"(1)] by blast
4718  AOT_hence ¬𝒜𝒜E!a using "&E"(2) "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by blast
4719  AOT_hence ¬𝒜E!a using "logic-actual-nec:4"[axiom_inst, THEN "≡E"(1)] RAA by blast
4720  AOT_thus p & ¬p for p using 1[THEN "&E"(1)] "&I" "raa-cor:1" by blast
4721qed
4722
4723AOT_theorem "pos-not-pna:2": ¬x(E!x & ¬𝒜E!x)
4724proof (rule RAA(1))
4725  AOT_show ¬𝒜x (E!x & ¬𝒜E!x) using "pos-not-pna:1" by blast
4726next
4727  AOT_assume ¬¬x (E!x & ¬𝒜E!x)
4728  AOT_hence x (E!x & ¬𝒜E!x)
4729    using "KBasic:12"[THEN "≡E"(2)] by blast
4730  AOT_thus 𝒜x (E!x & ¬𝒜E!x)
4731    using "nec-imp-act"[THEN "→E"] by blast
4732qed
4733
4734AOT_theorem "pos-not-pna:3": x (E!x & ¬𝒜E!x)
4735proof -
4736  AOT_obtain a where (E!a & ¬𝒜E!a)
4737    using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] "∃E"[rotated] by blast
4738  AOT_hence θ: E!a and ξ: ¬𝒜E!a using "KBasic2:3"[THEN "→E"] "&E" by blast+
4739  AOT_have ¬𝒜E!a using ξ "KBasic:11"[THEN "≡E"(2)] by blast
4740  AOT_hence ¬𝒜E!a using "Act-Basic:6"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)] by blast
4741  AOT_hence E!a & ¬𝒜E!a using θ "&I" by blast
4742  thus ?thesis using "∃I" by fast
4743qed
4744
4745AOT_define contingent_prop :: φ ("q0")
4746  q0_def: (q0) =df (x (E!x & ¬𝒜E!x))
4747
4748AOT_theorem q0_prop: q0 & ¬q0
4749  apply (rule "=dfI"(2)[OF q0_def])
4750  apply (fact "log-prop-prop:2")
4751  apply (rule "&I")
4752   apply (fact "qml:4"[axiom_inst])
4753  by (fact "pos-not-pna:2")
4754
4755AOT_theorem "basic-prop:1": Contingent0((q0))
4756proof(rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4757  AOT_have ¬Necessary0((q0)) & ¬Impossible0((q0))
4758  proof (rule "&I"; rule "=dfI"(2)[OF q0_def]; (rule "log-prop-prop:2" | rule "raa-cor:2"))
4759    AOT_assume Necessary0(x (E!x & ¬𝒜E!x))
4760    AOT_hence x (E!x & ¬𝒜E!x)
4761      using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4762    AOT_hence 𝒜x (E!x & ¬𝒜E!x)
4763      using "Act-Basic:8"[THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
4764    AOT_thus 𝒜x (E!x & ¬𝒜E!x) & ¬𝒜x (E!x & ¬𝒜E!x)
4765      using "pos-not-pna:1" "&I" by blast
4766  next
4767    AOT_assume Impossible0(x (E!x & ¬𝒜E!x))
4768    AOT_hence ¬(x (E!x & ¬𝒜E!x))
4769      using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4770    AOT_hence ¬(x (E!x & ¬𝒜E!x)) using "KBasic2:1"[THEN "≡E"(1)] by blast
4771    AOT_thus (x (E!x & ¬𝒜E!x)) & ¬(x (E!x & ¬𝒜E!x))
4772      using "qml:4"[axiom_inst] "&I" by blast
4773  qed
4774  AOT_thus ¬(Necessary0((q0))  Impossible0((q0)))
4775    using "oth-class-taut:5:d" "≡E"(2) by blast
4776qed
4777
4778AOT_theorem "basic-prop:2": p Contingent0((p))
4779  using "∃I"(1)[rotated, OF "log-prop-prop:2"] "basic-prop:1" by blast
4780
4781AOT_theorem "basic-prop:3": Contingent0(((q0)-))
4782  apply (AOT_subst ((q0)-) ¬q0)
4783   apply (insert "thm-relation-negation:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"]; fast)
4784  apply (rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4785  apply (rule "oth-class-taut:5:d"[THEN "≡E"(2)])
4786  apply (rule "&I")
4787   apply (rule "contingent-properties:1[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4788   apply (rule "conventions:5"[THEN "≡dfE"])
4789   apply (rule "=dfE"(2)[OF q0_def])
4790    apply (rule "log-prop-prop:2")
4791   apply (rule q0_prop[THEN "&E"(1)])
4792  apply (rule "contingent-properties:2[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4793  apply (rule "conventions:5"[THEN "≡dfE"])
4794  by (rule q0_prop[THEN "&E"(2)])
4795
4796AOT_theorem "basic-prop:4": pq (p  q & Contingent0(p) & Contingent0(q))
4797proof(rule "∃I")+
4798  AOT_have 0: φ  (φ)- for φ
4799    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4800  AOT_show (q0)  (q0)- & Contingent0(q0) & Contingent0(((q0)-))
4801    using "basic-prop:1" "basic-prop:3" "&I" 0 by presburger
4802qed(auto simp: "log-prop-prop:2")
4803
4804AOT_theorem "proposition-facts:1": NonContingent0(p)  ¬q (Contingent0(q) & q = p)
4805proof(rule "→I"; rule "raa-cor:2")
4806  AOT_assume NonContingent0(p)
4807  AOT_hence 1: Necessary0(p)  Impossible0(p)
4808    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4809  AOT_assume q (Contingent0(q) & q = p)
4810  then AOT_obtain q where Contingent0(q) & q = p using "∃E"[rotated] by blast
4811  AOT_hence Contingent0(p) using "rule=E" "&E" by fast
4812  AOT_thus (Necessary0(p)  Impossible0(p)) & ¬(Necessary0(p)  Impossible0(p))
4813    using "contingent-properties:4[zero]"[THEN "≡dfE"] 1 "&I" by blast
4814qed
4815
4816AOT_theorem "proposition-facts:2": Contingent0(p)  ¬q (NonContingent0(q) & q = p)
4817proof(rule "→I"; rule "raa-cor:2")
4818  AOT_assume Contingent0(p)
4819  AOT_hence 1: ¬(Necessary0(p)  Impossible0(p))
4820    using "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4821  AOT_assume q (NonContingent0(q) & q = p)
4822  then AOT_obtain q where NonContingent0(q) & q = p using "∃E"[rotated] by blast
4823  AOT_hence NonContingent0(p) using "rule=E" "&E" by fast
4824  AOT_thus (Necessary0(p)  Impossible0(p)) & ¬(Necessary0(p)  Impossible0(p))
4825    using "contingent-properties:3[zero]"[THEN "≡dfE"] 1 "&I" by blast
4826qed
4827
4828AOT_theorem "proposition-facts:3": (p0)  (p0)- & (p0)  (q0) & (p0)  (q0)- & (p0)-  (q0)- & (q0)  (q0)-
4829proof -
4830  {
4831    fix χ φ ψ
4832    AOT_assume χ{φ}
4833    moreover AOT_assume ¬χ{ψ}
4834    ultimately AOT_have ¬(χ{φ}  χ{ψ})
4835      using RAA "≡E" by metis
4836    moreover {
4837      AOT_have pq ((¬(χ{p}  χ{q}))  p  q)
4838        by (rule "∀I"; rule "∀I"; rule "pos-not-equiv-ne:4[zero]")
4839      AOT_hence ((¬(χ{φ}  χ{ψ}))  φ  ψ)
4840        using "∀E" "log-prop-prop:2" by blast
4841    }
4842    ultimately AOT_have φ  ψ
4843      using "→E" by blast
4844  } note 0 = this
4845  AOT_have contingent_neg: Contingent0(φ)  Contingent0(((φ)-)) for φ
4846    using "thm-cont-propos:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4847  AOT_have not_noncontingent_if_contingent: ¬NonContingent0(φ) if Contingent0(φ) for φ
4848    apply (rule "contingent-properties:3[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4849    using that "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4850  show ?thesis
4851    apply (rule "&I")+
4852    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4853       apply (rule 0)
4854    using "thm-noncont-propos:3" apply fast
4855       apply (rule not_noncontingent_if_contingent)
4856       apply (fact AOT)
4857      apply (rule 0)
4858    apply (rule "thm-noncont-propos:3")
4859      apply (rule not_noncontingent_if_contingent)
4860      apply (rule contingent_neg[THEN "≡E"(1)])
4861      apply (fact AOT)
4862     apply (rule 0)
4863    apply (rule "thm-noncont-propos:4")
4864      apply (rule not_noncontingent_if_contingent)
4865      apply (rule contingent_neg[THEN "≡E"(1)])
4866     apply (fact AOT)
4867    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4868qed
4869
4870AOT_define ContingentlyTrue :: ‹φ  φ› ("ContingentlyTrue'(_')")
4871  "cont-tf:1": ContingentlyTrue(p) df p & ¬p
4872
4873AOT_define ContingentlyFalse :: ‹φ  φ› ("ContingentlyFalse'(_')")
4874  "cont-tf:2": ContingentlyFalse(p) df ¬p & p
4875
4876AOT_theorem "cont-true-cont:1": ContingentlyTrue((p))  Contingent0((p))
4877proof(rule "→I")
4878  AOT_assume ContingentlyTrue((p))
4879  AOT_hence 1: p and 2: ¬p using "cont-tf:1"[THEN "≡dfE"] "&E" by blast+
4880  AOT_have ¬Necessary0((p))
4881    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4882    using 2 "KBasic:11"[THEN "≡E"(2)] by blast
4883  moreover AOT_have ¬Impossible0((p))
4884    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4885    apply (rule "conventions:5"[THEN "≡dfE"])
4886    using "T◇"[THEN "→E", OF 1].
4887  ultimately AOT_have ¬(Necessary0((p))  Impossible0((p)))
4888    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4889  AOT_thus Contingent0((p))
4890    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4891qed
4892
4893AOT_theorem "cont-true-cont:2": ContingentlyFalse((p))  Contingent0((p))
4894proof(rule "→I")
4895  AOT_assume ContingentlyFalse((p))
4896  AOT_hence 1: ¬p and 2: p using "cont-tf:2"[THEN "≡dfE"] "&E" by blast+
4897  AOT_have ¬Necessary0((p))
4898    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4899    using "KBasic:11"[THEN "≡E"(2)] "T◇"[THEN "→E", OF 1] by blast
4900  moreover AOT_have ¬Impossible0((p))
4901    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4902    apply (rule "conventions:5"[THEN "≡dfE"])
4903    using 2.
4904  ultimately AOT_have ¬(Necessary0((p))  Impossible0((p)))
4905    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4906  AOT_thus Contingent0((p))
4907    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4908qed
4909
4910AOT_theorem "cont-true-cont:3": ContingentlyTrue((p))  ContingentlyFalse(((p)-))
4911proof(rule "≡I"; rule "→I")
4912  AOT_assume ContingentlyTrue((p))
4913  AOT_hence 0: p & ¬p using "cont-tf:1"[THEN "≡dfE"] by blast
4914  AOT_have 1: ContingentlyFalse(¬p)
4915    apply (rule "cont-tf:2"[THEN "≡dfI"])
4916    apply (AOT_subst (reverse) ¬¬p p)
4917    by (auto simp: "oth-class-taut:3:b" 0)
4918  AOT_show ContingentlyFalse(((p)-))
4919    apply (AOT_subst ((p)-) ¬p)
4920    by (auto simp: "thm-relation-negation:3" 1)
4921next
4922  AOT_assume 1: ContingentlyFalse(((p)-))
4923  AOT_have ContingentlyFalse(¬p)
4924    by (AOT_subst (reverse) ¬p ((p)-))
4925       (auto simp: "thm-relation-negation:3" 1)
4926  AOT_hence ¬¬p & ¬p using "cont-tf:2"[THEN "≡dfE"] by blast
4927  AOT_hence p & ¬p
4928    using "&I" "&E" "useful-tautologies:1"[THEN "→E"] by metis
4929  AOT_thus ContingentlyTrue((p))
4930    using "cont-tf:1"[THEN "≡dfI"] by blast
4931qed
4932
4933AOT_theorem "cont-true-cont:4": ContingentlyFalse((p))  ContingentlyTrue(((p)-))
4934proof(rule "≡I"; rule "→I")
4935  AOT_assume ContingentlyFalse(p)
4936  AOT_hence 0: ¬p & p
4937    using "cont-tf:2"[THEN "≡dfE"] by blast
4938  AOT_have ¬p & ¬¬p
4939    by (AOT_subst (reverse) ¬¬p p)
4940       (auto simp: "oth-class-taut:3:b" 0)
4941  AOT_hence 1: ContingentlyTrue(¬p)
4942    by (rule "cont-tf:1"[THEN "≡dfI"])
4943  AOT_show ContingentlyTrue(((p)-))
4944    by (AOT_subst ((p)-) ¬p)
4945       (auto simp: "thm-relation-negation:3" 1)
4946next
4947  AOT_assume 1: ContingentlyTrue(((p)-))
4948  AOT_have ContingentlyTrue(¬p)
4949    by (AOT_subst (reverse) ¬p ((p)-))
4950       (auto simp add: "thm-relation-negation:3" 1)
4951  AOT_hence 2: ¬p & ¬¬p using "cont-tf:1"[THEN "≡dfE"] by blast
4952  AOT_have p
4953    by (AOT_subst p ¬¬p)
4954       (auto simp add: "oth-class-taut:3:b" 2[THEN "&E"(2)])
4955  AOT_hence ¬p & p using 2[THEN "&E"(1)] "&I" by blast
4956  AOT_thus ContingentlyFalse(p)
4957    by (rule "cont-tf:2"[THEN "≡dfI"])
4958qed
4959
4960AOT_theorem "cont-true-cont:5": (ContingentlyTrue((p)) & Necessary0((q)))  p  q
4961proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4962  AOT_assume ContingentlyTrue((p))
4963  AOT_hence ¬p
4964    using "cont-tf:1"[THEN "≡dfE"] "&E" by blast
4965  AOT_hence 0: ¬p using "KBasic:11"[THEN "≡E"(2)] by blast
4966  AOT_assume Necessary0((q))
4967  moreover AOT_assume ¬(p  q)
4968  AOT_hence p = q
4969    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4970          "useful-tautologies:1"[THEN "→E"] by blast
4971  ultimately AOT_have Necessary0((p)) using "rule=E" id_sym by blast
4972  AOT_hence p
4973    using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4974  AOT_thus p & ¬p using 0 "&I" by blast
4975qed
4976
4977AOT_theorem "cont-true-cont:6": (ContingentlyFalse((p)) & Impossible0((q)))  p  q
4978proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4979  AOT_assume ContingentlyFalse((p))
4980  AOT_hence p
4981    using "cont-tf:2"[THEN "≡dfE"] "&E" by blast
4982  AOT_hence 1: ¬¬p
4983    using "conventions:5"[THEN "≡dfE"] by blast
4984  AOT_assume Impossible0((q))
4985  moreover AOT_assume ¬(p  q)
4986  AOT_hence p = q
4987    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4988          "useful-tautologies:1"[THEN "→E"] by blast
4989  ultimately AOT_have Impossible0((p)) using "rule=E" id_sym by blast
4990  AOT_hence ¬p
4991    using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4992  AOT_thus ¬p & ¬¬p using 1 "&I" by blast
4993qed
4994
4995AOT_act_theorem "q0cf:1": ContingentlyFalse(q0)
4996  apply (rule "cont-tf:2"[THEN "≡dfI"])
4997  apply (rule "=dfI"(2)[OF q0_def])
4998   apply (fact "log-prop-prop:2")
4999  apply (rule "&I")
5000   apply (fact "no-cnac")
5001  by (fact "qml:4"[axiom_inst])
5002
5003AOT_act_theorem "q0cf:2": ContingentlyTrue(((q0)-))
5004  apply (rule "cont-tf:1"[THEN "≡dfI"])
5005  apply (rule "=dfI"(2)[OF q0_def])
5006   apply (fact "log-prop-prop:2")
5007  apply (rule "&I")
5008     apply (rule "thm-relation-negation:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(2)])
5009     apply (fact "no-cnac")
5010    apply (rule "rule=E"[rotated, OF "thm-relation-negation:7"[unvarify p, OF "log-prop-prop:2", THEN id_sym]])
5011  apply (AOT_subst (reverse) ¬¬(x  (E!x & ¬𝒜E!x)) x (E!x & ¬𝒜E!x))
5012  by (auto simp: "oth-class-taut:3:b" "qml:4"[axiom_inst])
5013
5014(* TODO: q0cf-rem skipped for now *)
5015
5016AOT_theorem "cont-tf-thm:1": p ContingentlyTrue((p))
5017proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
5018  AOT_assume q0
5019  AOT_hence q0 & ¬q0 using q0_prop[THEN "&E"(2)] "&I" by blast
5020  AOT_thus ContingentlyTrue(q0)
5021    by (rule "cont-tf:1"[THEN "≡dfI"])
5022next
5023  AOT_assume ¬q0
5024  AOT_hence ¬q0 & q0 using q0_prop[THEN "&E"(1)] "&I" by blast
5025  AOT_hence ContingentlyFalse(q0)
5026    by (rule "cont-tf:2"[THEN "≡dfI"])
5027  AOT_thus ContingentlyTrue(((q0)-))
5028    by (rule "cont-true-cont:4"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
5029qed(auto simp: "log-prop-prop:2")
5030
5031
5032AOT_theorem "cont-tf-thm:2": p ContingentlyFalse((p))
5033proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
5034  AOT_assume q0
5035  AOT_hence q0 & ¬q0 using q0_prop[THEN "&E"(2)] "&I" by blast
5036  AOT_hence ContingentlyTrue(q0)
5037    by (rule "cont-tf:1"[THEN "≡dfI"])
5038  AOT_thus ContingentlyFalse(((q0)-))
5039    by (rule "cont-true-cont:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
5040next
5041  AOT_assume ¬q0
5042  AOT_hence ¬q0 & q0 using q0_prop[THEN "&E"(1)] "&I" by blast
5043  AOT_thus ContingentlyFalse(q0)
5044    by (rule "cont-tf:2"[THEN "≡dfI"])
5045qed(auto simp: "log-prop-prop:2")
5046
5047(* TODO: inspect modally strict subproof involving obtained variable *)
5048AOT_theorem "property-facts1:1": Fx ([F]x & ¬[F]x)
5049proof -
5050  fix x
5051  AOT_obtain p1 where ContingentlyTrue((p1))
5052    using "cont-tf-thm:1" "∃E"[rotated] by blast
5053  AOT_hence 1: p1 & ¬p1 using "cont-tf:1"[THEN "≡dfE"] by blast
5054  AOT_modally_strict {
5055    AOT_have for arbitrary p:  (z p]x  p)
5056      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5057    AOT_hence for arbitrary p:   (z p]x  p)
5058      by (rule RN)
5059    AOT_hence p (z p]x  p) using GEN by fast
5060    AOT_hence (z p1]x  p1) using "∀E" by fast
5061  } note 2 = this
5062  AOT_hence (z p1]x  p1) using "∀E" by blast
5063  AOT_hence z p1]x using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
5064  moreover AOT_have ¬z p1]x
5065    apply (AOT_subst_using subst: 2[THEN "qml:2"[axiom_inst, THEN "→E"]])
5066    using 1[THEN "&E"(2)] by blast
5067  ultimately AOT_have z p1]x & ¬z p1]x using "&I" by blast
5068  AOT_hence x (z p1]x & ¬z p1]x) using "∃I"(2) by fast
5069  moreover AOT_have z p1] by "cqt:2[lambda]"
5070  ultimately AOT_show Fx ([F]x & ¬[F]x) by (rule "∃I"(1))
5071qed
5072
5073(* TODO: inspect modally strict subproof involving obtained variable *)
5074AOT_theorem "property-facts1:2": Fx (¬[F]x & [F]x)
5075proof -
5076  fix x
5077  AOT_obtain p1 where ContingentlyFalse((p1))
5078    using "cont-tf-thm:2" "∃E"[rotated] by blast
5079  AOT_hence 1: ¬p1 & p1 using "cont-tf:2"[THEN "≡dfE"] by blast
5080  AOT_modally_strict {
5081    AOT_have for arbitrary p:  (z p]x  p)
5082      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5083    AOT_hence for arbitrary p:  (¬z p]x  ¬p)
5084      using "oth-class-taut:4:b" "≡E" by blast
5085    AOT_hence for arbitrary p:  (¬z p]x  ¬p)
5086      by (rule RN)
5087    AOT_hence p (¬z p]x  ¬p) using GEN by fast
5088    AOT_hence (¬z p1]x  ¬p1) using "∀E" by fast
5089  } note 2 = this
5090  AOT_hence (¬z p1]x  ¬p1) using "∀E" by blast
5091  AOT_hence 3: ¬z p1]x using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
5092  AOT_modally_strict {
5093    AOT_have for arbitrary p:  (z p]x  p)
5094      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5095    AOT_hence for arbitrary p:  (z p]x  p)
5096      by (rule RN)
5097    AOT_hence p (z p]x  p) using GEN by fast
5098    AOT_hence (z p1]x  p1) using "∀E" by fast
5099  } note 4 = this
5100  AOT_have z p1]x
5101    apply (AOT_subst_using subst: 4[THEN "qml:2"[axiom_inst, THEN "→E"]])
5102    using 1[THEN "&E"(2)] by blast
5103  AOT_hence ¬z p1]x & z p1]x using 3 "&I" by blast
5104  AOT_hence x (¬z p1]x & z p1]x) using "∃I"(2) by fast
5105  moreover AOT_have z p1] by "cqt:2[lambda]"
5106  ultimately AOT_show Fx (¬[F]x & [F]x) by (rule "∃I"(1))
5107qed
5108
5109context
5110begin
5111
5112private AOT_lemma eqnotnec_123_Aux_ζ: [L]x  (E!x  E!x)
5113    apply (rule "=dfI"(2)[OF L_def])
5114     apply "cqt:2[lambda]"
5115    apply (rule "beta-C-meta"[THEN "→E"])
5116  by "cqt:2[lambda]"
5117
5118private AOT_lemma eqnotnec_123_Aux_ω: z φ]x  φ
5119    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5120
5121private AOT_lemma eqnotnec_123_Aux_θ: φ  x([L]x  z φ]x)
5122proof(rule "≡I"; rule "→I"; (rule "∀I")?)
5123  fix x
5124  AOT_assume 1: φ
5125  AOT_have [L]x  (E!x  E!x) using eqnotnec_123_Aux_ζ.
5126  also AOT_have   φ
5127    using "if-p-then-p" 1 "≡I" "→I" by simp
5128  also AOT_have   z φ]x
5129    using "Commutativity of ≡"[THEN "≡E"(1)] eqnotnec_123_Aux_ω by blast
5130  finally AOT_show [L]x  z φ]x.
5131next
5132  fix x
5133  AOT_assume x([L]x  z φ]x)
5134  AOT_hence [L]x  z φ]x using "∀E" by blast
5135  also AOT_have   φ using eqnotnec_123_Aux_ω.
5136  finally AOT_have φ  [L]x using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5137  also AOT_have   E!x  E!x using eqnotnec_123_Aux_ζ.
5138  finally AOT_show φ using "≡E" "if-p-then-p" by fast
5139qed
5140private lemmas eqnotnec_123_Aux_ξ =  eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5141                      THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5142                      THEN "RM◇"]
5143private lemmas eqnotnec_123_Aux_ξ' = eqnotnec_123_Aux_θ[THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)], THEN "RM◇"]
5144
5145AOT_theorem "eqnotnec:1": FG(x([F]x  [G]x) & ¬x([F]x  [G]x))
5146proof-
5147  AOT_obtain p1 where ContingentlyTrue(p1) using "cont-tf-thm:1" "∃E"[rotated] by blast
5148  AOT_hence p1 & ¬p1 using "cont-tf:1"[THEN "≡dfE"] by blast
5149  AOT_hence x ([L]x  z p1]x) & ¬x([L]x  z p1]x)
5150    apply - apply (rule "&I")
5151    using "&E" eqnotnec_123_Aux_θ[THEN "≡E"(1)] eqnotnec_123_Aux_ξ "→E" by fast+
5152  AOT_hence G (x([L]x  [G]x) & ¬x([L]x  [G]x))
5153    by (rule "∃I") "cqt:2[lambda]"
5154  AOT_thus FG (x([F]x  [G]x) & ¬x([F]x  [G]x))
5155    apply (rule "∃I")
5156    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5157qed
5158
5159AOT_theorem "eqnotnec:2": FG(¬x([F]x  [G]x) & x([F]x  [G]x))
5160proof-
5161  AOT_obtain p1 where ContingentlyFalse(p1) using "cont-tf-thm:2" "∃E"[rotated] by blast
5162  AOT_hence ¬p1 & p1 using "cont-tf:2"[THEN "≡dfE"] by blast
5163  AOT_hence ¬x ([L]x  z p1]x) & x([L]x  z p1]x)
5164    apply - apply (rule "&I")
5165    using "&E" eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)] eqnotnec_123_Aux_ξ' "→E" by fast+
5166  AOT_hence G (¬x([L]x  [G]x) & x([L]x  [G]x))
5167    by (rule "∃I") "cqt:2[lambda]"
5168  AOT_thus FG (¬x([F]x  [G]x) & x([F]x  [G]x))
5169    apply (rule "∃I")
5170    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5171qed
5172
5173AOT_theorem "eqnotnec:3": FG(𝒜¬x([F]x  [G]x) & x([F]x  [G]x))
5174proof-
5175  AOT_have ¬𝒜q0
5176    apply (rule "=dfI"(2)[OF q0_def])
5177     apply (fact "log-prop-prop:2")
5178    by (fact AOT)
5179  AOT_hence 𝒜¬q0
5180    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5181  AOT_hence 𝒜¬x ([L]x  z q0]x)
5182    using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5183            THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5184            THEN "RA[2]", THEN "act-cond"[THEN "→E"], THEN "→E"] by blast
5185  moreover AOT_have x ([L]x  z q0]x) using eqnotnec_123_Aux_ξ'[THEN "→E"] q0_prop[THEN "&E"(1)] by blast
5186  ultimately AOT_have 𝒜¬x ([L]x  z q0]x) & x ([L]x  z q0]x) using "&I" by blast
5187  AOT_hence G (𝒜¬x([L]x  [G]x) & x([L]x  [G]x))
5188    by (rule "∃I") "cqt:2[lambda]"
5189  AOT_thus FG (𝒜¬x([F]x  [G]x) & x([F]x  [G]x))
5190    apply (rule "∃I")
5191    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5192qed
5193
5194end
5195
5196(* TODO[IMPORTANT]: proof of 219.4 ζ: appeal to (159.2) requires a theorem, but the result has local
5197   assumptions! *)
5198AOT_theorem "eqnotnec:4": FG(x([F]x  [G]x) & ¬x([F]x  [G]x))
5199proof(rule GEN)
5200  fix F
5201
5202  AOT_have Aux_A:  ψ  x([F]x  z [F]z & ψ]x) for ψ
5203  proof(rule "→I"; rule GEN)
5204    AOT_modally_strict {
5205    fix x
5206    AOT_assume 0: ψ
5207    AOT_have z [F]z & ψ]x  [F]x & ψ
5208      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5209    also AOT_have ...  [F]x
5210      apply (rule "≡I"; rule "→I")
5211      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5212      using 0 "&I" by blast
5213    finally AOT_show [F]x  z [F]z & ψ]x
5214      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5215    }
5216  qed
5217
5218  AOT_have Aux_B:  ψ  x([F]x  z [F]z & ψ  ¬ψ]x) for ψ
5219  proof (rule "→I"; rule GEN)
5220    AOT_modally_strict {
5221      fix x
5222      AOT_assume 0: ψ
5223      AOT_have z ([F]z & ψ)  ¬ψ]x  (([F]x & ψ)  ¬ψ)
5224        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5225      also AOT_have ...  [F]x
5226        apply (rule "≡I"; rule "→I")
5227        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5228        apply (rule "∨I"(1)) using 0 "&I" by blast
5229      finally AOT_show [F]x  z ([F]z & ψ)  ¬ψ]x
5230        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5231    }
5232  qed
5233
5234  AOT_have Aux_C:  ¬ψ  ¬z(z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z) for ψ
5235  proof(rule "RM◇"; rule "→I"; rule "raa-cor:2")
5236  AOT_modally_strict {
5237      AOT_assume 0: ¬ψ
5238      AOT_assume z (z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z)
5239      AOT_hence z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5240      moreover AOT_have z [F]z & ψ]z  [F]z & ψ for z
5241          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5242      moreover AOT_have z ([F]z & ψ)  ¬ψ]z  (([F]z & ψ)  ¬ψ) for z
5243        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5244      ultimately AOT_have [F]z & ψ  (([F]z & ψ)  ¬ψ) for z
5245        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5246      moreover AOT_have (([F]z & ψ)  ¬ψ) for z using 0 "∨I" by blast
5247      ultimately AOT_have ψ using "≡E" "&E" by metis
5248      AOT_thus ψ & ¬ψ using 0 "&I" by blast
5249    }
5250  qed
5251
5252  AOT_have Aux_D: z ([F]z  z [F]z & ψ]z)  (¬x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)  ¬x ([F]x  z [F]z & ψ  ¬ψ]x)) for ψ
5253  proof (rule "→I")
5254    AOT_assume A: z([F]z  z [F]z & ψ]z)
5255    AOT_show ¬x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)  ¬x ([F]x  z [F]z & ψ  ¬ψ]x)
5256    proof(rule "≡I"; rule "KBasic:13"[THEN "→E"];
5257          rule "RN[prem]"[where Γ="{«z([F]z  z [F]z & ψ]z)»}", simplified];
5258          (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5259      AOT_modally_strict {
5260        AOT_assume z ([F]z  z [F]z & ψ]z)
5261        AOT_hence 1: [F]z  z [F]z & ψ]z for z using "∀E" by blast
5262        AOT_assume x ([F]x  z [F]z & ψ  ¬ψ]x)
5263        AOT_hence 2: [F]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5264        AOT_have z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "≡E" 1 2 by meson
5265        AOT_thus x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x) by (rule GEN)
5266      }
5267    next
5268      AOT_modally_strict {
5269        AOT_assume z ([F]z  z [F]z & ψ]z)
5270        AOT_hence 1: [F]z  z [F]z & ψ]z for z using "∀E" by blast
5271        AOT_assume x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)
5272        AOT_hence 2: z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5273        AOT_have [F]z  z [F]z & ψ  ¬ψ]z for z using 1 2 "≡E" by meson
5274        AOT_thus x ([F]x  z [F]z & ψ  ¬ψ]x) by (rule GEN)
5275      }
5276    qed(auto simp: A)
5277  qed
5278
5279  AOT_obtain p1 where p1_prop: p1 & ¬p1 using "cont-tf-thm:1" "∃E"[rotated] "cont-tf:1"[THEN "≡dfE"] by blast
5280  {
5281    AOT_assume 1: x([F]x  z [F]z & p1]x)
5282    AOT_have 2: x([F]x  z [F]z & p1  ¬p1]x)
5283      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5284    AOT_have ¬x(z [F]z & p1]x  z [F]z & p1  ¬p1]x)
5285      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5286    AOT_hence 3: ¬x([F]x  z [F]z & p1  ¬p1]x)
5287      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5288    AOT_hence x([F]x  z [F]z & p1  ¬p1]x) & ¬x([F]x  z [F]z & p1  ¬p1]x) using 2 "&I" by blast
5289    AOT_hence G (x ([F]x  [G]x) & ¬x([F]x  [G]x))
5290      by (rule "∃I"(1)) "cqt:2[lambda]"
5291  }
5292  moreover {
5293    AOT_assume 2: ¬x([F]x  z [F]z & p1]x)
5294    AOT_hence ¬x([F]x  z [F]z & p1]x)
5295      using "KBasic:11"[THEN "≡E"(1)] by blast
5296    AOT_hence x ([F]x  z [F]z & p1]x) & ¬x([F]x  z [F]z & p1]x)
5297      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(1)]] "&I" by blast
5298    AOT_hence G (x ([F]x  [G]x) & ¬x([F]x  [G]x))
5299      by (rule "∃I"(1)) "cqt:2[lambda]"
5300  }
5301  ultimately AOT_show G (x ([F]x  [G]x) & ¬x([F]x  [G]x))
5302    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5303qed
5304
5305AOT_theorem "eqnotnec:5": FG(¬x([F]x  [G]x) & x([F]x  [G]x))
5306proof(rule GEN)
5307  fix F
5308
5309  AOT_have Aux_A:  ψ  x([F]x  z [F]z & ψ]x) for ψ
5310  proof(rule "RM◇"; rule "→I"; rule GEN)
5311    AOT_modally_strict {
5312    fix x
5313    AOT_assume 0: ψ
5314    AOT_have z [F]z & ψ]x  [F]x & ψ
5315      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5316    also AOT_have ...  [F]x
5317      apply (rule "≡I"; rule "→I")
5318      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5319      using 0 "&I" by blast
5320    finally AOT_show [F]x  z [F]z & ψ]x
5321      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5322    }
5323  qed
5324
5325  AOT_have Aux_B:  ψ  x([F]x  z [F]z & ψ  ¬ψ]x) for ψ
5326  proof (rule "RM◇"; rule "→I"; rule GEN)
5327    AOT_modally_strict {
5328      fix x
5329      AOT_assume 0: ψ
5330      AOT_have z ([F]z & ψ)  ¬ψ]x  (([F]x & ψ)  ¬ψ)
5331        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5332      also AOT_have ...  [F]x
5333        apply (rule "≡I"; rule "→I")
5334        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5335        apply (rule "∨I"(1)) using 0 "&I" by blast
5336      finally AOT_show [F]x  z ([F]z & ψ)  ¬ψ]x
5337        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5338    }
5339  qed
5340
5341  AOT_have Aux_C:  ¬ψ  ¬z(z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z) for ψ
5342  proof(rule "→I"; rule "raa-cor:2")
5343  AOT_modally_strict {
5344      AOT_assume 0: ¬ψ
5345      AOT_assume z (z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z)
5346      AOT_hence z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5347      moreover AOT_have z [F]z & ψ]z  [F]z & ψ for z
5348          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5349      moreover AOT_have z ([F]z & ψ)  ¬ψ]z  (([F]z & ψ)  ¬ψ) for z
5350        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5351      ultimately AOT_have [F]z & ψ  (([F]z & ψ)  ¬ψ) for z
5352        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5353      moreover AOT_have (([F]z & ψ)  ¬ψ) for z using 0 "∨I" by blast
5354      ultimately AOT_have ψ using "≡E" "&E" by metis
5355      AOT_thus ψ & ¬ψ using 0 "&I" by blast
5356    }
5357  qed
5358
5359  AOT_have Aux_D: z ([F]z  z [F]z & ψ]z)  (¬x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)  ¬x ([F]x  z [F]z & ψ  ¬ψ]x)) for ψ
5360  proof (rule "→I"; rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5361    AOT_modally_strict {
5362      AOT_assume z ([F]z  z [F]z & ψ]z)
5363      AOT_hence 1: [F]z  z [F]z & ψ]z for z using "∀E" by blast
5364      AOT_assume x ([F]x  z [F]z & ψ  ¬ψ]x)
5365      AOT_hence 2: [F]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5366      AOT_have z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "≡E" 1 2 by meson
5367      AOT_thus x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x) by (rule GEN)
5368    }
5369  next
5370    AOT_modally_strict {
5371      AOT_assume z ([F]z  z [F]z & ψ]z)
5372      AOT_hence 1: [F]z  z [F]z & ψ]z for z using "∀E" by blast
5373      AOT_assume x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)
5374      AOT_hence 2: z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5375      AOT_have [F]z  z [F]z & ψ  ¬ψ]z for z using 1 2 "≡E" by meson
5376      AOT_thus x ([F]x  z [F]z & ψ  ¬ψ]x) by (rule GEN)
5377    }
5378  qed
5379
5380  AOT_obtain p1 where p1_prop: ¬p1 & p1 using "cont-tf-thm:2" "∃E"[rotated] "cont-tf:2"[THEN "≡dfE"] by blast
5381  {
5382    AOT_assume 1: x([F]x  z [F]z & p1]x)
5383    AOT_have 2: x([F]x  z [F]z & p1  ¬p1]x)
5384      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5385    AOT_have ¬x(z [F]z & p1]x  z [F]z & p1  ¬p1]x)
5386      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5387    AOT_hence 3: ¬x([F]x  z [F]z & p1  ¬p1]x)
5388      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5389    AOT_hence ¬x([F]x  z [F]z & p1  ¬p1]x) & x([F]x  z [F]z & p1  ¬p1]x) using 2 "&I" by blast
5390    AOT_hence G (¬x ([F]x  [G]x) & x([F]x  [G]x))
5391      by (rule "∃I"(1)) "cqt:2[lambda]"
5392  }
5393  moreover {
5394    AOT_assume 2: ¬x([F]x  z [F]z & p1]x)
5395    AOT_hence ¬x([F]x  z [F]z & p1]x)
5396      using "KBasic:11"[THEN "≡E"(1)] by blast
5397    AOT_hence ¬x ([F]x  z [F]z & p1]x) & x([F]x  z [F]z & p1]x)
5398      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(2)]] "&I" by blast
5399    AOT_hence G (¬x ([F]x  [G]x) & x([F]x  [G]x))
5400      by (rule "∃I"(1)) "cqt:2[lambda]"
5401  }
5402  ultimately AOT_show G (¬x ([F]x  [G]x) & x([F]x  [G]x))
5403    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5404qed
5405
5406AOT_theorem "eqnotnec:6": FG(𝒜¬x([F]x  [G]x) & x([F]x  [G]x))
5407proof(rule GEN)
5408  fix F
5409
5410  AOT_have Aux_A:  ψ  x([F]x  z [F]z & ψ]x) for ψ
5411  proof(rule "RM◇"; rule "→I"; rule GEN)
5412    AOT_modally_strict {
5413    fix x
5414    AOT_assume 0: ψ
5415    AOT_have z [F]z & ψ]x  [F]x & ψ
5416      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5417    also AOT_have ...  [F]x
5418      apply (rule "≡I"; rule "→I")
5419      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5420      using 0 "&I" by blast
5421    finally AOT_show [F]x  z [F]z & ψ]x
5422      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5423    }
5424  qed
5425
5426  AOT_have Aux_B:  ψ  x([F]x  z [F]z & ψ  ¬ψ]x) for ψ
5427  proof (rule "RM◇"; rule "→I"; rule GEN)
5428    AOT_modally_strict {
5429      fix x
5430      AOT_assume 0: ψ
5431      AOT_have z ([F]z & ψ)  ¬ψ]x  (([F]x & ψ)  ¬ψ)
5432        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5433      also AOT_have ...  [F]x
5434        apply (rule "≡I"; rule "→I")
5435        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5436        apply (rule "∨I"(1)) using 0 "&I" by blast
5437      finally AOT_show [F]x  z ([F]z & ψ)  ¬ψ]x
5438        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5439    }
5440  qed
5441
5442  AOT_have Aux_C:  𝒜¬ψ  𝒜¬z(z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z) for ψ
5443  proof(rule "act-cond"[THEN "→E"]; rule "RA[2]"; rule "→I"; rule "raa-cor:2")
5444  AOT_modally_strict {
5445      AOT_assume 0: ¬ψ
5446      AOT_assume z (z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z)
5447      AOT_hence z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5448      moreover AOT_have z [F]z & ψ]z  [F]z & ψ for z
5449          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5450      moreover AOT_have z ([F]z & ψ)  ¬ψ]z  (([F]z & ψ)  ¬ψ) for z
5451        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5452      ultimately AOT_have [F]z & ψ  (([F]z & ψ)  ¬ψ) for z
5453        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5454      moreover AOT_have (([F]z & ψ)  ¬ψ) for z using 0 "∨I" by blast
5455      ultimately AOT_have ψ using "≡E" "&E" by metis
5456      AOT_thus ψ & ¬ψ using 0 "&I" by blast
5457    }
5458  qed
5459
5460  AOT_have Aux_D: 𝒜z ([F]z  z [F]z & ψ]z)  (𝒜¬x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)  𝒜¬x ([F]x  z [F]z & ψ  ¬ψ]x)) for ψ
5461  proof (rule "→I"; rule "Act-Basic:5"[THEN "≡E"(1)])
5462    AOT_assume 𝒜z ([F]z  z [F]z & ψ]z)
5463    AOT_thus 𝒜(¬x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)  ¬x ([F]x  z [F]z & ψ  ¬ψ]x))
5464    proof (rule "RA[3]"[where Γ="{«z ([F]z  z [F]z & ψ]z)»}", simplified, rotated])
5465      AOT_modally_strict {
5466        AOT_assume z ([F]z  z [F]z & ψ]z)
5467        AOT_thus ¬x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)  ¬x ([F]x  z [F]z & ψ  ¬ψ]x)
5468          apply -
5469        proof(rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5470        AOT_modally_strict {
5471          AOT_assume z ([F]z  z [F]z & ψ]z)
5472          AOT_hence 1: [F]z  z [F]z & ψ]z for z using "∀E" by blast
5473          AOT_assume x ([F]x  z [F]z & ψ  ¬ψ]x)
5474          AOT_hence 2: [F]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5475          AOT_have z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "≡E" 1 2 by meson
5476          AOT_thus x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x) by (rule GEN)
5477        }
5478      next
5479        AOT_modally_strict {
5480          AOT_assume z ([F]z  z [F]z & ψ]z)
5481          AOT_hence 1: [F]z  z [F]z & ψ]z for z using "∀E" by blast
5482          AOT_assume x (z [F]z & ψ]x  z [F]z & ψ  ¬ψ]x)
5483          AOT_hence 2: z [F]z & ψ]z  z [F]z & ψ  ¬ψ]z for z using "∀E" by blast
5484          AOT_have [F]z  z [F]z & ψ  ¬ψ]z for z using 1 2 "≡E" by meson
5485          AOT_thus x ([F]x  z [F]z & ψ  ¬ψ]x) by (rule GEN)
5486        }
5487      qed
5488      }
5489    qed
5490  qed
5491
5492  AOT_have ¬𝒜q0
5493    apply (rule "=dfI"(2)[OF q0_def])
5494     apply (fact "log-prop-prop:2")
5495    by (fact AOT)
5496  AOT_hence q0_prop_1: 𝒜¬q0
5497    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5498  {
5499    AOT_assume 1: 𝒜x([F]x  z [F]z & q0]x)
5500    AOT_have 2: x([F]x  z [F]z & q0  ¬q0]x)
5501      using Aux_B[THEN "→E", OF q0_prop[THEN "&E"(1)]].
5502    AOT_have 𝒜¬x(z [F]z & q0]x  z [F]z & q0  ¬q0]x)
5503      using Aux_C[THEN "→E", OF q0_prop_1].
5504    AOT_hence 3: 𝒜¬x([F]x  z [F]z & q0  ¬q0]x)
5505      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5506    AOT_hence 𝒜¬x([F]x  z [F]z & q0  ¬q0]x) & x([F]x  z [F]z & q0  ¬q0]x) using 2 "&I" by blast
5507    AOT_hence G (𝒜¬x ([F]x  [G]x) & x([F]x  [G]x))
5508      by (rule "∃I"(1)) "cqt:2[lambda]"
5509  }
5510  moreover {
5511    AOT_assume 2: ¬𝒜x([F]x  z [F]z & q0]x)
5512    AOT_hence 𝒜¬x([F]x  z [F]z & q0]x)
5513      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5514    AOT_hence 𝒜¬x ([F]x  z [F]z & q0]x) & x([F]x  z [F]z & q0]x)
5515      using Aux_A[THEN "→E", OF q0_prop[THEN "&E"(1)]] "&I" by blast
5516    AOT_hence G (𝒜¬x ([F]x  [G]x) & x([F]x  [G]x))
5517      by (rule "∃I"(1)) "cqt:2[lambda]"
5518  }
5519  ultimately AOT_show G (𝒜¬x ([F]x  [G]x) & x([F]x  [G]x))
5520    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5521qed
5522
5523AOT_theorem "oa-contingent:1": O!  A!
5524proof(rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
5525  fix x
5526  AOT_assume 1: O! = A!
5527  AOT_hence x E!x] = A!
5528    by (rule "=dfE"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5529  AOT_hence x E!x] = x ¬E!x]
5530    by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5531  moreover AOT_have x E!x]x  E!x
5532    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5533  ultimately AOT_have x ¬E!x]x  E!x
5534    using "rule=E" by fast
5535  moreover AOT_have x ¬E!x]x  ¬E!x
5536    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5537  ultimately AOT_have E!x  ¬E!x using "≡E"(6) "Commutativity of ≡"[THEN "≡E"(1)] by blast
5538  AOT_thus "(E!x  ¬E!x) & ¬(E!x  ¬E!x)" using "oth-class-taut:3:c" "&I" by blast
5539qed
5540
5541AOT_theorem "oa-contingent:2": O!x  ¬A!x
5542proof -
5543  AOT_have O!x  x E!x]x
5544    apply (rule "≡I"; rule "→I")
5545     apply (rule "=dfE"(2)[OF AOT_ordinary])
5546      apply "cqt:2[lambda]"
5547     apply argo
5548    apply (rule  "=dfI"(2)[OF AOT_ordinary])
5549     apply "cqt:2[lambda]"
5550    by argo
5551  also AOT_have   E!x
5552    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5553  also AOT_have   ¬¬E!x
5554    using "oth-class-taut:3:b".
5555  also AOT_have   ¬x ¬E!x]x
5556    by (rule "beta-C-meta"[THEN "→E", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric]) "cqt:2[lambda]"
5557  also AOT_have   ¬A!x
5558    apply (rule "≡I"; rule "→I")
5559     apply (rule "=dfI"(2)[OF AOT_abstract])
5560      apply "cqt:2[lambda]"
5561     apply argo
5562    apply (rule "=dfE"(2)[OF AOT_abstract])
5563     apply "cqt:2[lambda]"
5564    by argo
5565  finally show ?thesis.
5566qed
5567
5568AOT_theorem "oa-contingent:3": A!x  ¬O!x
5569  by (AOT_subst A!x ¬¬A!x)
5570     (auto simp add: "oth-class-taut:3:b" "oa-contingent:2"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric])
5571
5572AOT_theorem "oa-contingent:4": Contingent(O!)
5573proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:1", THEN "≡E"(2)]; rule "&I")
5574  AOT_have x E!x using "thm-cont-e:3" .
5575  AOT_hence x E!x using "BF◇"[THEN "→E"] by blast
5576  then AOT_obtain a where E!a using "∃E"[rotated] by blast
5577  AOT_hence x E!x]a
5578    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5579  AOT_hence O!a
5580    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5581  AOT_hence x O!x using "∃I" by blast
5582  AOT_thus x O!x using "T◇"[THEN "→E"] by blast
5583next
5584  AOT_obtain a where A!a
5585    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5586  AOT_hence ¬O!a using "oa-contingent:3"[THEN "≡E"(1)] by blast
5587  AOT_hence x ¬O!x using "∃I" by fast
5588  AOT_thus x ¬O!x using "T◇"[THEN "→E"] by blast
5589qed
5590
5591AOT_theorem "oa-contingent:5": Contingent(A!)
5592proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:2", THEN "≡E"(2)]; rule "&I")
5593  AOT_obtain a where A!a
5594    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5595  AOT_hence x A!x using "∃I" by fast
5596  AOT_thus x A!x using "T◇"[THEN "→E"] by blast
5597next
5598  AOT_have x E!x using "thm-cont-e:3" .
5599  AOT_hence x E!x using "BF◇"[THEN "→E"] by blast
5600  then AOT_obtain a where E!a using "∃E"[rotated] by blast
5601  AOT_hence x E!x]a
5602    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5603  AOT_hence O!a
5604    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5605  AOT_hence ¬A!a using "oa-contingent:2"[THEN "≡E"(1)] by blast
5606  AOT_hence x ¬A!x using "∃I" by fast
5607  AOT_thus x ¬A!x using "T◇"[THEN "→E"] by blast
5608qed
5609
5610AOT_theorem "oa-contingent:7": O!-x  ¬A!-x
5611proof -
5612  AOT_have O!x  ¬A!x
5613    using "oa-contingent:2" by blast
5614  also AOT_have   A!-x
5615    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:2"].
5616  finally AOT_have 1: O!x  A!-x.
5617
5618  AOT_have A!x  ¬O!x
5619    using "oa-contingent:3" by blast
5620  also AOT_have   O!-x
5621    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:1"].
5622  finally AOT_have 2: A!x  O!-x.
5623
5624  AOT_show O!-x  ¬A!-x
5625    using 1[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "oa-contingent:3"[of _ x] 2[symmetric]
5626          "≡E"(5) by blast
5627qed
5628
5629AOT_theorem "oa-contingent:6": O!-  A!-
5630proof (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5631  AOT_assume 1: O!- = A!-
5632  fix x
5633  AOT_have A!-x  O!-x
5634    apply (rule "rule=E"[rotated, OF 1]) by (fact "oth-class-taut:3:a")
5635  AOT_hence A!-x  ¬A!-x
5636    using "oa-contingent:7" "≡E" by fast
5637  AOT_thus (A!-x  ¬A!-x) & ¬(A!-x  ¬A!-x) using "oth-class-taut:3:c" "&I" by blast
5638qed
5639
5640AOT_theorem "oa-contingent:8": Contingent(O!-)
5641  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:1", THEN "≡E"(1), OF "oa-contingent:4"].
5642
5643AOT_theorem "oa-contingent:9": Contingent(A!-)
5644  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:2", THEN "≡E"(1), OF "oa-contingent:5"].
5645
5646AOT_define WeaklyContingent :: ‹Π  φ› ("WeaklyContingent'(_')")
5647  "df-cont-nec": "WeaklyContingent([F]) df Contingent([F]) & x ([F]x  [F]x)"
5648
5649AOT_theorem "cont-nec-fact1:1": WeaklyContingent([F])  WeaklyContingent([F]-)
5650proof -
5651  AOT_have WeaklyContingent([F])  Contingent([F]) & x ([F]x  [F]x)
5652    using "df-cont-nec"[THEN "≡Df"] by blast
5653  also AOT_have ...  Contingent([F]-) & x ([F]x  [F]x)
5654    apply (rule "oth-class-taut:8:f"[THEN "≡E"(2)]; rule "→I")
5655    using "thm-cont-prop:3".
5656  also AOT_have   Contingent([F]-) & x ([F]-x  [F]-x)
5657  proof (rule "oth-class-taut:8:e"[THEN "≡E"(2)]; rule "→I"; rule "≡I"; rule "→I"; rule GEN; rule "→I")
5658    fix x
5659    AOT_assume 0: x ([F]x  [F]x)
5660    AOT_assume 1: [F]-x
5661    AOT_have ¬[F]x
5662      by (AOT_subst (reverse) ¬[F]x [F]-x)
5663         (auto simp add: "thm-relation-negation:1" 1)
5664    AOT_hence 2: ¬[F]x
5665      using "KBasic:11"[THEN "≡E"(2)] by blast
5666    AOT_show [F]-x
5667    proof (rule "raa-cor:1")
5668      AOT_assume 3: ¬[F]-x
5669      AOT_have ¬¬[F]x
5670        by (AOT_subst (reverse) ¬[F]x [F]-x)
5671           (auto simp add: "thm-relation-negation:1" 3)
5672      AOT_hence [F]x
5673        using "conventions:5"[THEN "≡dfI"] by simp
5674      AOT_hence [F]x using 0 "∀E" "→E" by fast
5675      AOT_thus [F]x & ¬[F]x using "&I" 2 by blast
5676    qed
5677  next
5678    fix x
5679    AOT_assume 0: x ([F]-x  [F]-x)
5680    AOT_assume 1: [F]x
5681    AOT_have ¬[F]-x
5682      by (AOT_subst ¬[F]-x [F]x)
5683         (auto simp: "thm-relation-negation:2" 1)
5684    AOT_hence 2: ¬[F]-x
5685      using "KBasic:11"[THEN "≡E"(2)] by blast
5686    AOT_show [F]x
5687    proof (rule "raa-cor:1")
5688      AOT_assume 3: ¬[F]x
5689      AOT_have ¬¬[F]-x
5690        by (AOT_subst ¬[F]-x [F]x)
5691           (auto simp add: "thm-relation-negation:2" 3)
5692      AOT_hence [F]-x
5693        using "conventions:5"[THEN "≡dfI"] by simp
5694      AOT_hence [F]-x using 0 "∀E" "→E" by fast
5695      AOT_thus [F]-x & ¬[F]-x using "&I" 2 by blast
5696    qed
5697  qed
5698  also AOT_have   WeaklyContingent([F]-)
5699    using "df-cont-nec"[THEN "≡Df", symmetric] by blast
5700  finally show ?thesis.
5701qed
5702
5703AOT_theorem "cont-nec-fact1:2": (WeaklyContingent([F]) & ¬WeaklyContingent([G]))  F  G
5704proof (rule "→I"; rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5705  AOT_assume 1: WeaklyContingent([F]) & ¬WeaklyContingent([G])
5706  AOT_hence WeaklyContingent([F]) using "&E" by blast
5707  moreover AOT_assume F = G
5708  ultimately AOT_have WeaklyContingent([G])
5709    using "rule=E" by blast
5710  AOT_thus WeaklyContingent([G]) & ¬WeaklyContingent([G])
5711    using 1 "&I" "&E" by blast
5712qed
5713
5714AOT_theorem "cont-nec-fact2:1": WeaklyContingent(O!)
5715proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5716  AOT_show Contingent(O!)
5717    using "oa-contingent:4".
5718next
5719  AOT_show x ([O!]x  [O!]x)
5720    apply (rule GEN; rule "→I")
5721    using "oa-facts:5"[THEN "≡E"(1)] by blast
5722qed
5723
5724
5725AOT_theorem "cont-nec-fact2:2": WeaklyContingent(A!)
5726proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5727  AOT_show Contingent(A!)
5728    using "oa-contingent:5".
5729next
5730  AOT_show x ([A!]x  [A!]x)
5731    apply (rule GEN; rule "→I")
5732    using "oa-facts:6"[THEN "≡E"(1)] by blast
5733qed
5734
5735AOT_theorem "cont-nec-fact2:3": ¬WeaklyContingent(E!)
5736proof (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5737       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "raa-cor:2")
5738  AOT_have x (E!x & ¬𝒜E!x) using "qml:4"[axiom_inst].
5739  AOT_hence x (E!x & ¬𝒜E!x) using "BF◇"[THEN "→E"] by blast
5740  then AOT_obtain a where (E!a & ¬𝒜E!a) using "∃E"[rotated] by blast
5741  AOT_hence 1: E!a & ¬𝒜E!a using "KBasic2:3"[THEN "→E"] by simp
5742  moreover AOT_assume x ([E!]x  [E!]x)
5743  ultimately AOT_have E!a using "&E" "∀E" "→E" by fast
5744  AOT_hence 𝒜E!a using "nec-imp-act"[THEN "→E"] by blast
5745  AOT_hence 𝒜E!a using "qml-act:1"[axiom_inst, THEN "→E"] by blast
5746  moreover AOT_have ¬𝒜E!a using "KBasic:11"[THEN "≡E"(2)] 1[THEN "&E"(2)] by meson
5747  ultimately AOT_have 𝒜E!a & ¬𝒜E!a using "&I" by blast
5748  AOT_thus p & ¬p for p using "raa-cor:1" by blast
5749qed
5750
5751AOT_theorem "cont-nec-fact2:4": ¬WeaklyContingent(L)
5752  apply (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5753       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(1))
5754  apply (rule "contingent-properties:4"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
5755  apply (rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "useful-tautologies:2"[THEN "→E"])
5756  using "thm-noncont-e-e:3"[THEN "contingent-properties:3"[THEN "≡dfE"]].
5757
5758(* TODO: cleanup *)
5759AOT_theorem "cont-nec-fact2:5": O!  E! & O!  E!- & O!  L & O!  L-
5760proof -
5761  AOT_have 1: L
5762    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5763  {
5764    fix φ and Π Π' :: <κ>
5765    AOT_have A: ¬(φ{Π'}  φ{Π}) if  φ{Π} and ¬φ{Π'}
5766    proof (rule "raa-cor:2")
5767      AOT_assume φ{Π'}  φ{Π}
5768      AOT_hence φ{Π'} using that(1) "≡E" by blast
5769      AOT_thus φ{Π'} & ¬φ{Π'} using that(2) "&I" by blast
5770    qed
5771    AOT_have Π'  Π if Π and Π' and φ{Π} and ¬φ{Π'}
5772      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5773  } note 0 = this
5774  show ?thesis
5775    apply(safe intro!: "&I"; rule 0)
5776    using "cqt:2[concrete]"[axiom_inst] apply blast
5777    using "oa-exist:1" apply blast
5778    using "cont-nec-fact2:3" apply fast
5779    apply (rule "useful-tautologies:2"[THEN "→E"])
5780    using "cont-nec-fact2:1" apply fast
5781    using "rel-neg-T:3" apply fast
5782    using "oa-exist:1" apply blast
5783    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5784    apply (rule "useful-tautologies:2"[THEN "→E"])
5785    using "cont-nec-fact2:1" apply blast
5786    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5787    using "oa-exist:1" apply fast
5788    using "cont-nec-fact2:4" apply fast
5789    apply (rule "useful-tautologies:2"[THEN "→E"])
5790    using "cont-nec-fact2:1" apply fast
5791    using "rel-neg-T:3" apply fast
5792    using "oa-exist:1" apply fast
5793    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5794    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5795    apply (rule "useful-tautologies:2"[THEN "→E"])
5796    using "cont-nec-fact2:1" by blast
5797qed
5798
5799(* TODO: cleanup together with above *)
5800AOT_theorem "cont-nec-fact2:6": A!  E! & A!  E!- & A!  L & A!  L-
5801proof -
5802  AOT_have 1: L
5803    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5804  {
5805    fix φ and Π Π' :: <κ>
5806    AOT_have A: ¬(φ{Π'}  φ{Π}) if  φ{Π} and ¬φ{Π'}
5807    proof (rule "raa-cor:2")
5808      AOT_assume φ{Π'}  φ{Π}
5809      AOT_hence φ{Π'} using that(1) "≡E" by blast
5810      AOT_thus φ{Π'} & ¬φ{Π'} using that(2) "&I" by blast
5811    qed
5812    AOT_have Π'  Π if Π and Π' and φ{Π} and ¬φ{Π'}
5813      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5814  } note 0 = this
5815  show ?thesis
5816    apply(safe intro!: "&I"; rule 0)
5817    using "cqt:2[concrete]"[axiom_inst] apply blast
5818    using "oa-exist:2" apply blast
5819    using "cont-nec-fact2:3" apply fast
5820    apply (rule "useful-tautologies:2"[THEN "→E"])
5821    using "cont-nec-fact2:2" apply fast
5822    using "rel-neg-T:3" apply fast
5823    using "oa-exist:2" apply blast
5824    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5825    apply (rule "useful-tautologies:2"[THEN "→E"])
5826    using "cont-nec-fact2:2" apply blast
5827    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5828    using "oa-exist:2" apply fast
5829    using "cont-nec-fact2:4" apply fast
5830    apply (rule "useful-tautologies:2"[THEN "→E"])
5831    using "cont-nec-fact2:2" apply fast
5832    using "rel-neg-T:3" apply fast
5833    using "oa-exist:2" apply fast
5834    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5835    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5836    apply (rule "useful-tautologies:2"[THEN "→E"])
5837    using "cont-nec-fact2:2" by blast
5838qed
5839
5840AOT_define necessary_or_contingently_false :: ‹φ  φ› ("Δ_" [49] 54)
5841  Δp df p  (¬𝒜p & p)
5842
5843AOT_theorem sixteen:
5844 shows F1F2F3F4F5F6F7F8F9F10F11F12F13F14F15F16 (
5845«F1::<κ>»  F2 & F1  F3 & F1  F4 & F1  F5 & F1  F6 & F1  F7 & F1  F8 & F1  F9 & F1  F10 & F1  F11 & F1  F12 & F1  F13 & F1  F14 & F1  F15 & F1  F16 &
5846F2  F3 & F2  F4 & F2  F5 & F2  F6 & F2  F7 & F2  F8 & F2  F9 & F2  F10 & F2  F11 & F2  F12 & F2  F13 & F2  F14 & F2  F15 & F2  F16 &
5847F3  F4 & F3  F5 & F3  F6 & F3  F7 & F3  F8 & F3  F9 & F3  F10 & F3  F11 & F3  F12 & F3  F13 & F3  F14 & F3  F15 & F3  F16 &
5848F4  F5 & F4  F6 & F4  F7 & F4  F8 & F4  F9 & F4  F10 & F4  F11 & F4  F12 & F4  F13 & F4  F14 & F4  F15 & F4  F16 &
5849F5  F6 & F5  F7 & F5  F8 & F5  F9 & F5  F10 & F5  F11 & F5  F12 & F5  F13 & F5  F14 & F5  F15 & F5  F16 &
5850F6  F7 & F6  F8 & F6  F9 & F6  F10 & F6  F11 & F6  F12 & F6  F13 & F6  F14 & F6  F15 & F6  F16 &
5851F7  F8 & F7  F9 & F7  F10 & F7  F11 & F7  F12 & F7  F13 & F7  F14 & F7  F15 & F7  F16 &
5852F8  F9 & F8  F10 & F8  F11 & F8  F12 & F8  F13 & F8  F14 & F8  F15 & F8  F16 &
5853F9  F10 & F9  F11 & F9  F12 & F9  F13 & F9  F14 & F9  F15 & F9  F16 &
5854F10  F11 & F10  F12 & F10  F13 & F10  F14 & F10  F15 & F10  F16 &
5855F11  F12 & F11  F13 & F11  F14 & F11  F15 & F11  F16 &
5856F12  F13 & F12  F14 & F12  F15 & F12  F16 &
5857F13  F14 & F13  F15 & F13  F16 &
5858F14  F15 & F14  F16 &
5859F15  F16) 
5860proof -
5861
5862  AOT_have Delta_pos: Δφ  φ for φ
5863  proof(rule "→I")
5864    AOT_assume Δφ
5865    AOT_hence φ  (¬𝒜φ & φ)
5866      using "≡dfE"[OF necessary_or_contingently_false] by blast
5867    moreover {
5868      AOT_assume φ
5869      AOT_hence φ
5870        by (metis "B◇" "T◇" "vdash-properties:10")
5871    }
5872    moreover {
5873      AOT_assume ¬𝒜φ & φ
5874      AOT_hence φ
5875        using "&E" by blast
5876    }
5877    ultimately AOT_show φ
5878      by (metis "∨E"(2) "raa-cor:1") 
5879  qed
5880
5881  AOT_have act_and_not_nec_not_delta: ¬Δφ if 𝒜φ and ¬φ for φ
5882    using "≡dfE" "&E"(1) "∨E"(2) necessary_or_contingently_false "raa-cor:3" that(1) that(2) by blast
5883  AOT_have act_and_pos_not_not_delta: ¬Δφ if 𝒜φ and ¬φ for φ
5884    using "KBasic:11" act_and_not_nec_not_delta "≡E"(2) that(1) that(2) by blast
5885  AOT_have impossible_delta: ¬Δφ if ¬φ for φ
5886    using Delta_pos "modus-tollens:1" that by blast
5887  AOT_have not_act_and_pos_delta: Δφ if ¬𝒜φ and φ for φ
5888    by (meson "≡dfI" "&I" "∨I"(2) necessary_or_contingently_false that(1) that(2))
5889  AOT_have nec_delta: Δφ if φ for φ
5890    using "≡dfI" "∨I"(1) necessary_or_contingently_false that by blast
5891
5892  AOT_obtain a where a_prop: A!a
5893    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5894  AOT_obtain b where b_prop: [E!]b & ¬𝒜[E!]b
5895    using "pos-not-pna:3" using "∃E"[rotated] by blast
5896
5897  AOT_have b_ord: [O!]b
5898  proof(rule "=dfI"(2)[OF AOT_ordinary])
5899    AOT_show x [E!]x] by "cqt:2[lambda]"
5900  next
5901    AOT_show x [E!]x]b
5902    proof (rule "β←C"(1); ("cqt:2[lambda]")?)
5903      AOT_show b by (rule "cqt:2[const_var]"[axiom_inst])
5904      AOT_show [E!]b by (fact b_prop[THEN "&E"(1)])
5905    qed
5906  qed
5907
5908  AOT_have nec_not_L_neg: ¬[L-]x for x
5909    using "thm-noncont-e-e:2" "contingent-properties:2"[THEN "≡dfE"] "&E"
5910          CBF[THEN "→E"] "∀E" by blast
5911  AOT_have nec_L: [L]x for x
5912    using "thm-noncont-e-e:1" "contingent-properties:1"[THEN "≡dfE"]
5913      CBF[THEN "→E"] "∀E" by blast
5914
5915  AOT_have act_ord_b: 𝒜[O!]b
5916    using b_ord "≡E"(1) "oa-facts:7" by blast
5917  AOT_have delta_ord_b: Δ[O!]b
5918    by (meson "≡dfI" b_ord "∨I"(1) necessary_or_contingently_false "oa-facts:1" "vdash-properties:10")
5919  AOT_have not_act_ord_a: ¬𝒜[O!]a
5920    by (meson a_prop "≡E"(1) "≡E"(3) "oa-contingent:3" "oa-facts:7")
5921  AOT_have not_delta_ord_a: ¬Δ[O!]a
5922    by (metis Delta_pos "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "reductio-aa:1" "vdash-properties:10")
5923
5924  AOT_have not_act_abs_b: ¬𝒜[A!]b
5925    by (meson b_ord "≡E"(1) "≡E"(3) "oa-contingent:2" "oa-facts:8")
5926  AOT_have not_delta_abs_b: ¬Δ[A!]b
5927  proof(rule "raa-cor:2")
5928    AOT_assume Δ[A!]b
5929    AOT_hence [A!]b
5930      by (metis Delta_pos "vdash-properties:10")
5931    AOT_thus [A!]b & ¬[A!]b
5932      by (metis b_ord "&I" "≡E"(1) "oa-contingent:2" "oa-facts:4" "vdash-properties:10")
5933  qed
5934  AOT_have act_abs_a: 𝒜[A!]a
5935    using a_prop "≡E"(1) "oa-facts:8" by blast
5936  AOT_have delta_abs_a: Δ[A!]a
5937      by (metis "≡dfI" a_prop "oa-facts:2" "vdash-properties:10" "∨I"(1) necessary_or_contingently_false)
5938
5939  AOT_have not_act_concrete_b: ¬𝒜[E!]b
5940    using b_prop "&E"(2) by blast
5941  AOT_have delta_concrete_b: Δ[E!]b
5942  proof (rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5943    AOT_show ¬𝒜[E!]b using b_prop "&E"(2) by blast
5944  next
5945    AOT_show [E!]b using b_prop "&E"(1) by blast
5946  qed
5947  AOT_have not_act_concrete_a: ¬𝒜[E!]a
5948  proof (rule "raa-cor:2")
5949    AOT_assume 𝒜[E!]a
5950    AOT_hence 1: [E!]a by (metis "Act-Sub:3" "vdash-properties:10")
5951    AOT_have [A!]a by (simp add: a_prop)
5952    AOT_hence x ¬[E!]x]a
5953      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5954    AOT_hence ¬[E!]a using "β→C"(1) by blast
5955    AOT_thus [E!]a & ¬[E!]a using 1 "&I" by blast
5956  qed
5957  AOT_have not_delta_concrete_a: ¬Δ[E!]a
5958  proof (rule "raa-cor:2")
5959    AOT_assume Δ[E!]a
5960    AOT_hence 1: [E!]a by (metis Delta_pos "vdash-properties:10")
5961    AOT_have [A!]a by (simp add: a_prop)
5962    AOT_hence x ¬[E!]x]a
5963      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5964    AOT_hence ¬[E!]a using "β→C"(1) by blast
5965    AOT_thus [E!]a & ¬[E!]a using 1 "&I" by blast
5966  qed
5967
5968  AOT_have not_act_q_zero: ¬𝒜q0
5969    by (meson "log-prop-prop:2" "pos-not-pna:1" q0_def "reductio-aa:1" "rule-id-df:2:a[zero]")
5970  AOT_have delta_q_zero: Δq0
5971  proof(rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5972    AOT_show ¬𝒜q0 using not_act_q_zero.
5973    AOT_show q0 by (meson "&E"(1) q0_prop)
5974  qed
5975  AOT_have act_not_q_zero: 𝒜¬q0 using "Act-Basic:1" "∨E"(2) not_act_q_zero by blast
5976  AOT_have not_delta_not_q_zero: ¬Δ¬q0
5977      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta "&E"(1) "∨E"(2) not_act_q_zero q0_prop by blast
5978
5979  AOT_have [L-] by (simp add: "rel-neg-T:3")
5980  moreover AOT_have ¬𝒜[L-]b & ¬Δ[L-]b & ¬𝒜[L-]a & ¬Δ[L-]a
5981  proof (safe intro!: "&I")
5982    AOT_show ¬𝒜[L-]b by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5983    AOT_show ¬Δ[L-]b by (meson Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg)
5984    AOT_show ¬𝒜[L-]a by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5985    AOT_show ¬Δ[L-]a using Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg by blast
5986  qed
5987  ultimately AOT_obtain F0 where ¬𝒜[F0]b & ¬Δ[F0]b & ¬𝒜[F0]a & ¬Δ[F0]a
5988    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5989  AOT_hence ¬𝒜[F0]b and ¬Δ[F0]b and ¬𝒜[F0]a and ¬Δ[F0]a
5990    using "&E" by blast+
5991  note props = this
5992
5993  let  = "«y [A!]y & q0]»"
5994  AOT_modally_strict {
5995    AOT_have [«»] by "cqt:2[lambda]"
5996  } note 1 = this
5997  moreover AOT_have¬𝒜[«»]b & ¬Δ[«»]b & ¬𝒜[«»]a & Δ[«»]a
5998  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5999    AOT_show ¬𝒜([A!]b & q0)
6000      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
6001  next AOT_show ¬Δ([A!]b & q0)
6002      by (metis Delta_pos "KBasic2:3" "&E"(1) "≡E"(4) not_act_abs_b "oa-facts:4" "oa-facts:8" "raa-cor:3" "vdash-properties:10")
6003  next AOT_show ¬𝒜([A!]a & q0)
6004      using "Act-Basic:2" "&E"(2) "≡E"(1) not_act_q_zero "raa-cor:3" by blast
6005  next AOT_show Δ([A!]a & q0)
6006    proof (rule not_act_and_pos_delta)
6007      AOT_show ¬𝒜([A!]a & q0)
6008        using "Act-Basic:2" "&E"(2) "≡E"(4) not_act_q_zero "raa-cor:3" by blast
6009    next AOT_show ([A!]a & q0)
6010        by (metis "&I" "→E" Delta_pos "KBasic:16" "&E"(1) delta_abs_a "≡E"(1) "oa-facts:6" q0_prop)
6011    qed
6012  qed
6013  ultimately AOT_obtain F1 where ¬𝒜[F1]b & ¬Δ[F1]b & ¬𝒜[F1]a & Δ[F1]a
6014    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6015  AOT_hence ¬𝒜[F1]b and ¬Δ[F1]b and ¬𝒜[F1]a and Δ[F1]a
6016    using "&E" by blast+
6017  note props = props this
6018
6019  let  = "«y [A!]y & ¬q0]»"
6020  AOT_modally_strict {
6021    AOT_have [«»] by "cqt:2[lambda]"
6022  } note 1 = this
6023  moreover AOT_have ¬𝒜[«»]b & ¬Δ[«»]b & 𝒜[«»]a & ¬Δ[«»]a
6024  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6025    AOT_show ¬𝒜([A!]b & ¬q0)
6026      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
6027  next AOT_show ¬Δ([A!]b & ¬q0)
6028      by (meson "RM◇" Delta_pos "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b "oa-facts:4" "oa-facts:8")
6029  next AOT_show 𝒜([A!]a & ¬q0)
6030      by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
6031  next AOT_show ¬Δ([A!]a & ¬q0)
6032    proof (rule act_and_not_nec_not_delta)
6033      AOT_show 𝒜([A!]a & ¬q0)
6034        by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
6035    next
6036      AOT_show ¬([A!]a & ¬q0)
6037        by (metis "KBasic2:1" "KBasic:3" "&E"(1) "&E"(2) "≡E"(4) q0_prop "raa-cor:3")
6038    qed
6039  qed
6040  ultimately AOT_obtain F2 where ¬𝒜[F2]b & ¬Δ[F2]b & 𝒜[F2]a & ¬Δ[F2]a
6041    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6042  AOT_hence ¬𝒜[F2]b and ¬Δ[F2]b and 𝒜[F2]a and ¬Δ[F2]a
6043    using "&E" by blast+
6044  note props = props this
6045
6046  AOT_have abstract_prop: ¬𝒜[A!]b & ¬Δ[A!]b & 𝒜[A!]a & Δ[A!]a
6047    using act_abs_a "&I" delta_abs_a not_act_abs_b not_delta_abs_b by presburger
6048  then AOT_obtain F3 where ¬𝒜[F3]b & ¬Δ[F3]b & 𝒜[F3]a & Δ[F3]a
6049    using "∃I"(1)[rotated, THEN "∃E"[rotated]] "oa-exist:2" by fastforce
6050  AOT_hence ¬𝒜[F3]b and ¬Δ[F3]b and 𝒜[F3]a and Δ[F3]a
6051    using "&E" by blast+
6052  note props = props this
6053
6054  AOT_have ¬𝒜[E!]b & Δ[E!]b & ¬𝒜[E!]a & ¬Δ[E!]a
6055    by (meson "&I" delta_concrete_b not_act_concrete_a not_act_concrete_b not_delta_concrete_a)
6056  then AOT_obtain F4 where ¬𝒜[F4]b & Δ[F4]b & ¬𝒜[F4]a & ¬Δ[F4]a
6057    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6058  AOT_hence ¬𝒜[F4]b and Δ[F4]b and ¬𝒜[F4]a and ¬Δ[F4]a
6059    using "&E" by blast+
6060  note props = props this
6061
6062  AOT_modally_strict {
6063    AOT_have y q0] by "cqt:2[lambda]"
6064  } note 1 = this
6065  moreover AOT_have ¬𝒜y q0]b & Δy q0]b & ¬𝒜y q0]a & Δy q0]a
6066    by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6067       (auto simp: not_act_q_zero delta_q_zero)
6068  ultimately AOT_obtain F5 where ¬𝒜[F5]b & Δ[F5]b & ¬𝒜[F5]a & Δ[F5]a
6069    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6070  AOT_hence ¬𝒜[F5]b and Δ[F5]b and ¬𝒜[F5]a and Δ[F5]a
6071    using "&E" by blast+
6072  note props = props this
6073
6074  let  = "«y [E!]y  ([A!]y & ¬q0)]»"
6075  AOT_modally_strict {
6076    AOT_have [«»] by "cqt:2[lambda]"
6077  } note 1 = this
6078  moreover AOT_have ¬𝒜[«»]b & Δ[«»]b & 𝒜[«»]a & ¬Δ[«»]a
6079  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6080    AOT_have 𝒜¬([A!]b & ¬q0)
6081      by (metis "Act-Basic:1" "Act-Basic:2" abstract_prop "&E"(1) "∨E"(2)
6082                "≡E"(1) "raa-cor:3")
6083    moreover AOT_have ¬𝒜[E!]b
6084      using b_prop "&E"(2) by blast
6085    ultimately AOT_have 2: 𝒜(¬[E!]b & ¬([A!]b & ¬q0))
6086      by (metis "Act-Basic:2" "Act-Sub:1" "&I" "≡E"(3) "raa-cor:1")
6087    AOT_have 𝒜¬([E!]b  ([A!]b & ¬q0))
6088      by (AOT_subst ¬([E!]b  ([A!]b & ¬q0)) ¬[E!]b & ¬([A!]b & ¬q0))
6089         (auto simp: "oth-class-taut:5:d" 2)
6090    AOT_thus ¬𝒜([E!]b  ([A!]b & ¬q0))
6091      by (metis "¬¬I" "Act-Sub:1" "≡E"(4))
6092  next
6093    AOT_show Δ([E!]b  ([A!]b & ¬q0))
6094    proof (rule not_act_and_pos_delta)
6095      AOT_show ¬𝒜([E!]b  ([A!]b & ¬q0))
6096        by (metis "Act-Basic:2" "Act-Basic:9" "∨E"(2) "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b not_act_concrete_b "raa-cor:3")
6097    next
6098      AOT_show ([E!]b  ([A!]b & ¬q0))
6099        using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6100    qed
6101  next AOT_show 𝒜([E!]a  ([A!]a & ¬q0))
6102      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6103  next AOT_show ¬Δ([E!]a  ([A!]a & ¬q0))
6104    proof (rule act_and_not_nec_not_delta)
6105      AOT_show 𝒜([E!]a  ([A!]a & ¬q0))
6106        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6107    next
6108      AOT_have ¬[E!]a
6109        by (metis "≡dfI" "conventions:5" "&I" "∨I"(2) necessary_or_contingently_false not_act_concrete_a not_delta_concrete_a "raa-cor:3")
6110      moreover AOT_have ¬([A!]a & ¬q0)
6111        by (metis "KBasic2:1" "KBasic:11" "KBasic:3" "&E"(1) "&E"(2) "≡E"(1) q0_prop "raa-cor:3")
6112      ultimately AOT_have (¬[E!]a & ¬([A!]a & ¬q0)) by (metis "KBasic:16" "&I" "vdash-properties:10")
6113      AOT_hence ¬([E!]a  ([A!]a & ¬q0))
6114        by (metis "RE◇" "≡E"(2) "oth-class-taut:5:d")
6115      AOT_thus ¬([E!]a  ([A!]a & ¬q0)) by (metis "KBasic:12" "≡E"(1) "raa-cor:3")
6116    qed
6117  qed
6118  ultimately AOT_obtain F6 where ¬𝒜[F6]b & Δ[F6]b & 𝒜[F6]a & ¬Δ[F6]a
6119    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6120  AOT_hence ¬𝒜[F6]b and Δ[F6]b and 𝒜[F6]a and ¬Δ[F6]a
6121    using "&E" by blast+
6122  note props = props this
6123
6124  let  = "«y [A!]y  [E!]y]»"
6125  AOT_modally_strict {
6126    AOT_have [«»] by "cqt:2[lambda]"
6127  } note 1 = this
6128  moreover AOT_have ¬𝒜[«»]b & Δ[«»]b & 𝒜[«»]a & Δ[«»]a
6129  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6130    AOT_show ¬𝒜([A!]b  [E!]b)
6131      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6132  next AOT_show Δ([A!]b  [E!]b)
6133    proof (rule not_act_and_pos_delta)
6134      AOT_show ¬𝒜([A!]b  [E!]b)
6135        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6136    next AOT_show ([A!]b  [E!]b)
6137        using "KBasic2:2" b_prop "&E"(1) "∨I"(2) "≡E"(2) by blast
6138    qed
6139  next AOT_show 𝒜([A!]a  [E!]a)
6140      by (meson "Act-Basic:9" act_abs_a "∨I"(1) "≡E"(2))
6141  next AOT_show Δ([A!]a  [E!]a)
6142    proof (rule nec_delta)
6143      AOT_show ([A!]a  [E!]a)
6144        by (metis "KBasic:15" act_abs_a act_and_not_nec_not_delta "Disjunction Addition"(1) delta_abs_a "raa-cor:3" "vdash-properties:10")
6145    qed
6146  qed
6147  ultimately AOT_obtain F7 where ¬𝒜[F7]b & Δ[F7]b & 𝒜[F7]a & Δ[F7]a
6148    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6149  AOT_hence ¬𝒜[F7]b and Δ[F7]b and 𝒜[F7]a and Δ[F7]a
6150    using "&E" by blast+
6151  note props = props this
6152
6153  let  = "«y [O!]y & ¬[E!]y]»"
6154  AOT_modally_strict {
6155    AOT_have [«»] by "cqt:2[lambda]"
6156  } note 1 = this
6157  moreover AOT_have 𝒜[«»]b & ¬Δ[«»]b & ¬𝒜[«»]a & ¬Δ[«»]a
6158  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6159    AOT_show 𝒜([O!]b & ¬[E!]b)
6160      by (metis "Act-Basic:1" "Act-Basic:2" act_ord_b "&I" "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:3")
6161  next AOT_show ¬Δ([O!]b & ¬[E!]b)
6162      by (metis (no_types, hide_lams) "conventions:5" "Act-Sub:1" "RM:1" act_and_not_nec_not_delta "act-conj-act:3"
6163                act_ord_b b_prop "&I" "&E"(1) "Conjunction Simplification"(2) "df-rules-formulas[3]"
6164                "≡E"(3) "raa-cor:1" "→E")
6165  next AOT_show ¬𝒜([O!]a & ¬[E!]a)
6166      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_ord_a "raa-cor:3" by blast
6167  next AOT_have ¬([O!]a & ¬[E!]a)
6168      by (metis "KBasic2:3" "&E"(1) "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "raa-cor:3" "vdash-properties:10")
6169    AOT_thus ¬Δ([O!]a & ¬[E!]a)
6170      by (rule impossible_delta)
6171  qed      
6172  ultimately AOT_obtain F8 where 𝒜[F8]b & ¬Δ[F8]b & ¬𝒜[F8]a & ¬Δ[F8]a
6173    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6174  AOT_hence 𝒜[F8]b and ¬Δ[F8]b and ¬𝒜[F8]a and ¬Δ[F8]a
6175    using "&E" by blast+
6176  note props = props this
6177
6178  (* TODO_PLM: binary property 9 wrong in PLM *)
6179  let  = "«y ¬[E!]y & ([O!]y  q0)]»"
6180  AOT_modally_strict {
6181    AOT_have [«»] by "cqt:2[lambda]"
6182  } note 1 = this
6183  moreover AOT_have 𝒜[«»]b & ¬Δ[«»]b & ¬𝒜[«»]a & Δ[«»]a
6184  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6185    AOT_show 𝒜(¬[E!]b & ([O!]b  q0))
6186      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6187                "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6188  next AOT_show ¬Δ(¬[E!]b & ([O!]b  q0))
6189    proof (rule act_and_pos_not_not_delta)
6190      AOT_show 𝒜(¬[E!]b & ([O!]b  q0))
6191        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6192                  "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6193    next
6194      AOT_show ¬(¬[E!]b & ([O!]b  q0))
6195      proof (AOT_subst ¬(¬[E!]b & ([O!]b  q0)) [E!]b  ¬([O!]b  q0))
6196        AOT_modally_strict {
6197          AOT_show ¬(¬[E!]b & ([O!]b  q0))  [E!]b  ¬([O!]b  q0)
6198            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2) "∨E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
6199        }
6200      next
6201        AOT_show ([E!]b  ¬([O!]b  q0))
6202          using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6203       qed
6204     qed
6205   next
6206     AOT_show ¬𝒜(¬[E!]a & ([O!]a  q0))
6207       using "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2" by blast
6208   next
6209     AOT_show Δ(¬[E!]a & ([O!]a  q0))
6210     proof (rule not_act_and_pos_delta)
6211       AOT_show ¬𝒜(¬[E!]a & ([O!]a  q0))
6212         by (metis "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2")
6213     next
6214       AOT_have ¬[E!]a
6215         using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "raa-cor:5" by blast
6216       moreover AOT_have ([O!]a  q0)
6217         by (metis "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(3) q0_prop "raa-cor:3")
6218       ultimately AOT_show (¬[E!]a & ([O!]a  q0))
6219         by (metis "KBasic:16" "&I" "vdash-properties:10")
6220     qed
6221   qed
6222  ultimately AOT_obtain F9 where 𝒜[F9]b & ¬Δ[F9]b & ¬𝒜[F9]a & Δ[F9]a
6223    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6224  AOT_hence 𝒜[F9]b and ¬Δ[F9]b and ¬𝒜[F9]a and Δ[F9]a
6225    using "&E" by blast+
6226  note props = props this
6227
6228  AOT_modally_strict {
6229    AOT_have y ¬q0] by "cqt:2[lambda]"
6230  } note 1 = this
6231  moreover AOT_have 𝒜y ¬q0]b & ¬Δy ¬q0]b & 𝒜y ¬q0]a & ¬Δy ¬q0]a
6232    by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1]; auto simp: act_not_q_zero not_delta_not_q_zero)
6233  ultimately AOT_obtain F10 where 𝒜[F10]b & ¬Δ[F10]b & 𝒜[F10]a & ¬Δ[F10]a
6234    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6235  AOT_hence 𝒜[F10]b and ¬Δ[F10]b and 𝒜[F10]a and ¬Δ[F10]a
6236    using "&E" by blast+
6237  note props = props this
6238
6239  AOT_modally_strict {
6240    AOT_have y ¬[E!]y] by "cqt:2[lambda]"
6241  } note 1 = this
6242  moreover AOT_have 𝒜y ¬[E!]y]b & ¬Δy ¬[E!]y]b & 𝒜y ¬[E!]y]a & Δy ¬[E!]y]a
6243  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6244    AOT_show 𝒜¬[E!]b
6245      using "Act-Basic:1" "∨E"(2) not_act_concrete_b by blast
6246  next AOT_show ¬Δ¬[E!]b
6247      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta b_prop "&E"(1) "∨E"(2) not_act_concrete_b by blast
6248  next AOT_show 𝒜¬[E!]a
6249      using "Act-Basic:1" "∨E"(2) not_act_concrete_a by blast
6250  next AOT_show Δ¬[E!]a
6251      using "KBasic2:1" "≡E"(2) nec_delta not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "reductio-aa:1" by blast
6252  qed
6253  ultimately AOT_obtain F11 where 𝒜[F11]b & ¬Δ[F11]b & 𝒜[F11]a & Δ[F11]a
6254    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6255  AOT_hence 𝒜[F11]b and ¬Δ[F11]b and 𝒜[F11]a and Δ[F11]a
6256    using "&E" by blast+
6257  note props = props this
6258
6259  AOT_have 𝒜[O!]b & Δ[O!]b & ¬𝒜[O!]a & ¬Δ[O!]a
6260    by (simp add: act_ord_b "&I" delta_ord_b not_act_ord_a not_delta_ord_a)
6261  then AOT_obtain F12 where 𝒜[F12]b & Δ[F12]b & ¬𝒜[F12]a & ¬Δ[F12]a
6262    using "oa-exist:1" "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6263  AOT_hence 𝒜[F12]b and Δ[F12]b and ¬𝒜[F12]a and ¬Δ[F12]a
6264    using "&E" by blast+
6265  note props = props this
6266
6267  let  = "«y [O!]y  q0]»"
6268  AOT_modally_strict {
6269    AOT_have [«»] by "cqt:2[lambda]"
6270  } note 1 = this
6271  moreover AOT_have 𝒜[«»]b & Δ[«»]b & ¬𝒜[«»]a & Δ[«»]a
6272  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6273    AOT_show 𝒜([O!]b  q0)
6274      by (meson "Act-Basic:9" act_ord_b "∨I"(1) "≡E"(2))
6275  next AOT_show Δ([O!]b  q0)
6276      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6277  next AOT_show ¬𝒜([O!]a  q0)
6278      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6279  next AOT_show Δ([O!]a  q0)
6280    proof (rule not_act_and_pos_delta)
6281      AOT_show ¬𝒜([O!]a  q0)
6282        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6283    next AOT_show ([O!]a  q0)
6284        using "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(2) q0_prop by blast
6285    qed
6286  qed
6287  ultimately AOT_obtain F13 where 𝒜[F13]b & Δ[F13]b & ¬𝒜[F13]a & Δ[F13]a
6288    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6289  AOT_hence 𝒜[F13]b and Δ[F13]b and ¬𝒜[F13]a and Δ[F13]a
6290    using "&E" by blast+
6291  note props = props this
6292
6293  let  = "«y [O!]y  ¬q0]»"
6294  AOT_modally_strict {
6295     AOT_have [«»] by "cqt:2[lambda]"
6296  } note 1 = this
6297  moreover AOT_have 𝒜[«»]b & Δ[«»]b & 𝒜[«»]a & ¬Δ[«»]a
6298  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6299    AOT_show 𝒜([O!]b  ¬q0)
6300      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6301  next AOT_show Δ([O!]b  ¬q0)
6302      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6303  next AOT_show 𝒜([O!]a  ¬q0)
6304      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6305  next AOT_show ¬Δ([O!]a  ¬q0)
6306    proof(rule act_and_pos_not_not_delta)
6307      AOT_show 𝒜([O!]a  ¬q0)
6308        by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6309    next
6310      AOT_have ¬[O!]a
6311        using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_ord_a not_delta_ord_a "raa-cor:6" by blast
6312      moreover AOT_have q0
6313        by (meson "&E"(1) q0_prop)
6314      ultimately AOT_have 2: (¬[O!]a & q0)
6315         by (metis "KBasic:16" "&I" "vdash-properties:10")
6316      AOT_show ¬([O!]a  ¬q0)
6317      proof (AOT_subst (reverse) ¬([O!]a  ¬q0) ¬[O!]a & q0)
6318        AOT_modally_strict {
6319          AOT_show ¬[O!]a & q0  ¬([O!]a  ¬q0)
6320            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2)
6321                      "∨E"(3) "deduction-theorem" "≡I" "raa-cor:3")
6322        }
6323      next
6324        AOT_show (¬[O!]a & q0)
6325          using "2" by blast
6326      qed
6327    qed
6328  qed
6329  ultimately AOT_obtain F14 where 𝒜[F14]b & Δ[F14]b & 𝒜[F14]a & ¬Δ[F14]a
6330    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6331  AOT_hence 𝒜[F14]b and Δ[F14]b and 𝒜[F14]a and ¬Δ[F14]a
6332    using "&E" by blast+
6333  note props = props this
6334
6335  AOT_have [L]
6336    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
6337  moreover AOT_have 𝒜[L]b & Δ[L]b & 𝒜[L]a & Δ[L]a
6338  proof (safe intro!: "&I")
6339    AOT_show 𝒜[L]b
6340      by (meson nec_L "nec-imp-act" "vdash-properties:10")
6341    next AOT_show Δ[L]b using nec_L nec_delta by blast
6342    next AOT_show 𝒜[L]a by (meson nec_L "nec-imp-act" "vdash-properties:10")
6343    next AOT_show Δ[L]a using nec_L nec_delta by blast
6344  qed
6345  ultimately AOT_obtain F15 where 𝒜[F15]b & Δ[F15]b & 𝒜[F15]a & Δ[F15]a
6346    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6347  AOT_hence 𝒜[F15]b and Δ[F15]b and 𝒜[F15]a and Δ[F15]a
6348    using "&E" by blast+
6349  note props = props this
6350
6351  show ?thesis
6352    by (rule "∃I"(2)[where β=F0]; rule "∃I"(2)[where β=F1]; rule "∃I"(2)[where β=F2];
6353           rule "∃I"(2)[where β=F3]; rule "∃I"(2)[where β=F4]; rule "∃I"(2)[where β=F5];
6354           rule "∃I"(2)[where β=F6]; rule "∃I"(2)[where β=F7]; rule "∃I"(2)[where β=F8];
6355           rule "∃I"(2)[where β=F9]; rule "∃I"(2)[where β=F10]; rule "∃I"(2)[where β=F11];
6356           rule "∃I"(2)[where β=F12]; rule "∃I"(2)[where β=F13]; rule "∃I"(2)[where β=F14];
6357           rule "∃I"(2)[where β=F15]; safe intro!: "&I")
6358       (match conclusion in "[?v  [F]  [G]]" for F G  6359        match props in A: "[?v  ¬φ{F}]" for φ 6360        match (φ) in "λa . ?p" fail¦ "λa . a" fail¦ _ 6361        match props in B: "[?v  φ{G}]" 6362        fact "pos-not-equiv-ne:4"[where F=F and G=G and φ=φ, THEN "→E",
6363                                OF "oth-class-taut:4:h"[THEN "≡E"(2)],
6364                                OF "Disjunction Addition"(2)[THEN "→E"],
6365                                OF "&I", OF A, OF B]››››)+
6366qed
6367
6368AOT_theorem "o-objects-exist:1": x O!x
6369proof(rule RN)
6370  AOT_modally_strict {
6371    AOT_obtain a where (E!a & ¬𝒜[E!]a)
6372      using "∃E"[rotated, OF "qml:4"[axiom_inst, THEN "BF◇"[THEN "→E"]]] by blast
6373    AOT_hence 1: E!a by (metis "KBasic2:3" "&E"(1) "→E")
6374    AOT_have x [E!]x]a
6375    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6376      AOT_show a using "cqt:2[const_var]"[axiom_inst] by blast
6377    next
6378      AOT_show E!a by (fact 1)
6379    qed
6380    AOT_hence O!a by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6381    AOT_thus x [O!]x by (rule "∃I")
6382  }
6383qed
6384
6385AOT_theorem "o-objects-exist:2": x A!x
6386proof (rule RN)
6387  AOT_modally_strict {
6388    AOT_obtain a where [A!]a
6389      using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
6390    AOT_thus x A!x using "∃I" by blast
6391  }
6392qed
6393
6394AOT_theorem "o-objects-exist:3": ¬x O!x
6395  by (rule RN) (metis (no_types, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(4) "modus-tollens:1" "o-objects-exist:2" "oa-contingent:2" "qml:2"[axiom_inst] "reductio-aa:2")
6396
6397AOT_theorem "o-objects-exist:4": ¬x A!x
6398  by (rule RN) (metis (mono_tags, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(1) "modus-tollens:1" "o-objects-exist:1" "oa-contingent:2" "qml:2"[axiom_inst] "→E")
6399
6400AOT_theorem "o-objects-exist:5": ¬x E!x
6401proof (rule RN; rule "raa-cor:2")
6402  AOT_modally_strict {
6403    AOT_assume x E!x
6404    moreover AOT_obtain a where abs: A!a
6405      using "o-objects-exist:2"[THEN "qml:2"[axiom_inst, THEN "→E"]] "∃E"[rotated] by blast
6406    ultimately AOT_have E!a using "∀E" by blast
6407    AOT_hence 1: E!a by (metis "T◇" "→E")
6408    AOT_have y E!y]a
6409    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6410      AOT_show a using "cqt:2[const_var]"[axiom_inst].
6411    next
6412      AOT_show E!a by (fact 1)
6413    qed
6414    AOT_hence O!a
6415      by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6416    AOT_hence ¬A!a by (metis "≡E"(1) "oa-contingent:2") 
6417    AOT_thus p & ¬p for p using abs by (metis "raa-cor:3")
6418  }
6419qed
6420
6421AOT_theorem partition: ¬x (O!x & A!x)
6422proof(rule "raa-cor:2")
6423  AOT_assume x (O!x & A!x)
6424  then AOT_obtain a where O!a & A!a using "∃E"[rotated] by blast
6425  AOT_thus p & ¬p for p by (metis "&E"(1) "Conjunction Simplification"(2) "≡E"(1) "modus-tollens:1" "oa-contingent:2" "raa-cor:3")
6426qed
6427
6428AOT_define eq_E :: ‹Π› ("'(=E')") "=E": (=E) =df xy O!x & O!y & F ([F]x  [F]y)]
6429
6430syntax "_AOT_eq_E_infix" :: ‹τ  τ  φ› (infixl "=E" 50)
6431translations
6432  "_AOT_eq_E_infix κ κ'" == "CONST AOT_exe (CONST eq_E) (CONST Pair κ κ')"
6433(* TODO: try to replace by a simple translations pattern *)
6434print_translation6435AOT_syntax_print_translations
6436[(const_syntax‹AOT_exe›, fn ctxt => fn [
6437  Const ("constAOT_PLM.eq_E", _),
6438  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6439] => Const (syntax_const‹_AOT_eq_E_infix›, dummyT) $ lhs $ rhs)]
6440
6441text‹Note: Not explicitly mentioned as theorem in PLM.›
6442AOT_theorem "=E[denotes]": [(=E)]
6443  by (rule "=dfI"(2)[OF "=E"]) "cqt:2[lambda]"+
6444
6445AOT_theorem "=E-simple:1": x =E y  (O!x & O!y & F ([F]x  [F]y))
6446proof -
6447  (* TODO: rethink the product hacks *)
6448  AOT_have 0: «(AOT_term_of_var x,AOT_term_of_var y)»
6449    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6450  AOT_have 1: xy [O!]x & [O!]y & F ([F]x  [F]y)] by "cqt:2[lambda]"
6451  show ?thesis apply (rule "=dfI"(2)[OF "=E"]; "cqt:2[lambda]"?)
6452    using "beta-C-meta"[THEN "→E", OF 1, unvarify ν1νn, of "(AOT_term_of_var x,AOT_term_of_var y)", OF 0]
6453    by fast
6454qed
6455
6456AOT_theorem "=E-simple:2": x =E y  x = y
6457proof (rule "→I")
6458  AOT_assume x =E y
6459  AOT_hence O!x & O!y & F ([F]x  [F]y) using "=E-simple:1"[THEN "≡E"(1)] by blast
6460  AOT_thus x = y
6461    using "≡dfI"[OF "identity:1"] "∨I" by blast
6462qed
6463
6464AOT_theorem "id-nec3:1": x =E y  (x =E y)
6465proof (rule "≡I"; rule "→I")
6466  AOT_assume x =E y
6467  AOT_hence O!x & O!y & F ([F]x  [F]y)
6468    using "=E-simple:1" "≡E" by blast
6469  AOT_hence O!x & O!y & F ([F]x  [F]y)
6470    by (metis "S5Basic:6" "&I" "&E"(1) "&E"(2) "≡E"(4) "oa-facts:1" "raa-cor:3" "vdash-properties:10")
6471  AOT_hence (O!x & O!y & F ([F]x  [F]y))
6472    by (metis "&E"(1) "&E"(2) "≡E"(2) "KBasic:3" "&I")
6473  AOT_thus (x =E y)
6474    using "=E-simple:1"
6475    by (AOT_subst x =E y O!x & O!y & F ([F]x  [F]y)) auto
6476next
6477  AOT_assume (x =E y)
6478  AOT_thus x =E y using "qml:2"[axiom_inst, THEN "→E"] by blast
6479qed
6480
6481AOT_theorem "id-nec3:2": (x =E y)  x =E y
6482  by (meson "RE◇" "S5Basic:2" "id-nec3:1" "≡E"(1) "≡E"(5) "Commutativity of ≡")
6483
6484AOT_theorem "id-nec3:3": (x =E y)  (x =E y)
6485  by (meson "id-nec3:1" "id-nec3:2" "≡E"(5))
6486
6487syntax "_AOT_non_eq_E" :: ‹Π› ("'(≠E')")
6488translations
6489  (Π) "(≠E)" == (Π) "(=E)-"
6490syntax "_AOT_non_eq_E_infix" :: ‹τ  τ  φ› (infixl "E" 50)
6491translations
6492 "_AOT_non_eq_E_infix κ κ'" == "CONST AOT_exe (CONST relation_negation (CONST eq_E)) (CONST Pair κ κ')"
6493(* TODO: try replacing be a simple translations pattern *)
6494print_translation6495AOT_syntax_print_translations
6496[(const_syntax‹AOT_exe›, fn ctxt => fn [
6497  Const (const_syntax‹relation_negation›, _) $ Const ("constAOT_PLM.eq_E", _),
6498  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6499] => Const (syntax_const‹_AOT_non_eq_E_infix›, dummyT) $ lhs $ rhs)]
6500AOT_theorem "thm-neg=E": x E y  ¬(x =E y)
6501proof -
6502  (* TODO: rethink the product hacks *)
6503  AOT_have 0: «(AOT_term_of_var x,AOT_term_of_var y)»
6504    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6505  AOT_have θ: x1...x2 ¬(=E)x1...x2] by "cqt:2[lambda]" (* TODO_PLM: convoluted proof in PLM; TODO: product hack *)
6506  AOT_have x E y  x1...x2 ¬(=E)x1...x2]xy
6507    by (rule "=dfI"(1)[OF "df-relation-negation", OF θ])
6508       (meson "oth-class-taut:3:a")
6509  also AOT_have   ¬(=E)xy
6510    apply (rule "beta-C-meta"[THEN "→E", unvarify ν1νn])
6511     apply "cqt:2[lambda]"
6512    by (fact 0)
6513  finally show ?thesis.
6514qed
6515
6516AOT_theorem "id-nec4:1": x E y  (x E y)
6517proof -
6518  AOT_have x E y  ¬(x =E y) using "thm-neg=E".
6519  also AOT_have   ¬(x =E y)
6520    by (meson "id-nec3:2" "≡E"(1) "Commutativity of ≡" "oth-class-taut:4:b")
6521  also AOT_have   ¬(x =E y)
6522    by (meson "KBasic2:1" "≡E"(2) "Commutativity of ≡")
6523  also AOT_have   (x E y)
6524    by (AOT_subst (reverse) ¬(x =E y) x E y)
6525       (auto simp: "thm-neg=E" "oth-class-taut:3:a")
6526  finally show ?thesis.
6527qed
6528
6529AOT_theorem "id-nec4:2": (x E y)  (x E y)
6530  by (meson "RE◇" "S5Basic:2" "id-nec4:1" "≡E"(2) "≡E"(5) "Commutativity of ≡")
6531
6532AOT_theorem "id-nec4:3": (x E y)  (x E y)
6533  by (meson "id-nec4:1" "id-nec4:2" "≡E"(5))
6534
6535AOT_theorem "id-act2:1": x =E y  𝒜x =E y
6536  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec3:2" "≡E"(1) "≡E"(6))
6537AOT_theorem "id-act2:2": x E y  𝒜x E y
6538  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec4:2" "≡E"(1) "≡E"(6))
6539
6540AOT_theorem "ord=Eequiv:1": O!x  x =E x
6541proof (rule "→I")
6542  AOT_assume 1: O!x
6543  AOT_show x =E x
6544    apply (rule "=dfI"(2)[OF "=E"]) apply "cqt:2[lambda]"
6545    apply (rule "β←C"(1))
6546      apply "cqt:2[lambda]"
6547     apply (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6548    by (simp add: "1" RN "&I" "oth-class-taut:3:a" "universal-cor")
6549qed
6550
6551AOT_theorem "ord=Eequiv:2": x =E y  y =E x
6552proof(rule CP)
6553  AOT_assume 1: x =E y
6554  AOT_hence 2: x = y by (metis "=E-simple:2" "vdash-properties:10") 
6555  AOT_have O!x using 1 by (meson "&E"(1) "=E-simple:1" "≡E"(1))
6556  AOT_hence x =E x using "ord=Eequiv:1" "→E" by blast
6557  AOT_thus y =E x using "rule=E"[rotated, OF 2] by fast
6558qed
6559
6560AOT_theorem "ord=Eequiv:3": (x =E y & y =E z)  x =E z
6561proof (rule CP)
6562  AOT_assume 1: x =E y & y =E z
6563  AOT_hence x = y & y = z
6564    by (metis "&I" "&E"(1) "&E"(2) "=E-simple:2" "vdash-properties:6")
6565  AOT_hence x = z by (metis "id-eq:3" "vdash-properties:6")
6566  moreover AOT_have x =E x
6567    using 1[THEN "&E"(1)] "&E"(1) "=E-simple:1" "≡E"(1) "ord=Eequiv:1" "→E" by blast
6568  ultimately AOT_show x =E z
6569    using "rule=E" by fast
6570qed
6571
6572AOT_theorem "ord-=E=:1": (O!x  O!y)  (x = y  x =E y)
6573proof(rule CP)
6574  AOT_assume O!x  O!y
6575  moreover {
6576    AOT_assume O!x
6577    AOT_hence O!x by (metis "oa-facts:1" "vdash-properties:10")
6578    moreover {
6579      AOT_modally_strict {
6580        AOT_have O!x  (x = y  x =E y)
6581        proof (rule "→I"; rule "≡I"; rule "→I")
6582          AOT_assume O!x
6583          AOT_hence x =E x by (metis "ord=Eequiv:1" "→E")
6584          moreover AOT_assume x = y
6585          ultimately AOT_show x =E y using "rule=E" by fast
6586        next
6587          AOT_assume x =E y
6588          AOT_thus x = y by (metis "=E-simple:2" "→E")
6589        qed
6590      }
6591      AOT_hence O!x  (x = y  x =E y) by (metis "RM:1")
6592    }
6593    ultimately AOT_have (x = y  x =E y) using "→E" by blast
6594  }
6595  moreover {
6596    AOT_assume O!y
6597    AOT_hence O!y by (metis "oa-facts:1" "vdash-properties:10")
6598    moreover {
6599      AOT_modally_strict {
6600        AOT_have O!y  (x = y  x =E y)
6601        proof (rule "→I"; rule "≡I"; rule "→I")
6602          AOT_assume O!y
6603          AOT_hence y =E y by (metis "ord=Eequiv:1" "→E")
6604          moreover AOT_assume x = y
6605          ultimately AOT_show x =E y using "rule=E" id_sym by fast
6606        next
6607          AOT_assume x =E y
6608          AOT_thus x = y by (metis "=E-simple:2" "→E")
6609        qed
6610      }
6611      AOT_hence O!y  (x = y  x =E y) by (metis "RM:1")
6612    }
6613    ultimately AOT_have (x = y  x =E y) using "→E" by blast
6614  }
6615  ultimately AOT_show (x = y  x =E y) by (metis "∨E"(3) "raa-cor:1")
6616qed
6617
6618AOT_theorem "ord-=E=:2": O!y  x x = y]
6619proof (rule "→I"; rule "safe-ext"[axiom_inst, THEN "→E"]; rule "&I")
6620  AOT_show x x =E y] by "cqt:2[lambda]"
6621next
6622  AOT_assume O!y
6623  AOT_hence 1: (x = y  x =E y) for x using "ord-=E=:1" "→E" "∨I" by blast
6624  AOT_have (x =E y  x = y) for x
6625    by (AOT_subst x =E y  x = y x = y  x =E y)
6626       (auto simp add: "Commutativity of ≡" 1)
6627  AOT_hence x (x =E y  x = y) by (rule GEN)
6628  AOT_thus x (x =E y  x = y) by (rule BF[THEN "→E"])
6629qed
6630
6631
6632AOT_theorem "ord-=E=:3": xy O!x & O!y & x = y]
6633proof (rule "safe-ext[2]"[axiom_inst, THEN "→E"]; rule "&I")
6634  AOT_show xy O!x & O!y & x =E y] by "cqt:2[lambda]"
6635next
6636  AOT_show xy ([O!]x & [O!]y & x =E y  [O!]x & [O!]y & x = y)
6637  proof (rule RN; rule GEN; rule GEN; rule "≡I"; rule "→I")
6638    AOT_modally_strict {
6639      AOT_show [O!]x & [O!]y & x = y if [O!]x & [O!]y & x =E y for x y
6640        by (metis "&I" "&E"(1) "Conjunction Simplification"(2) "=E-simple:2"
6641                  "modus-tollens:1" "raa-cor:1" that)
6642    }
6643  next
6644    AOT_modally_strict {
6645      AOT_show [O!]x & [O!]y & x =E y if [O!]x & [O!]y & x = y for x y
6646        apply(safe intro!: "&I")
6647          apply (metis that[THEN "&E"(1), THEN "&E"(1)])
6648         apply (metis that[THEN "&E"(1), THEN "&E"(2)])
6649        using "rule=E"[rotated, OF that[THEN "&E"(2)]]
6650              "ord=Eequiv:1"[THEN "→E", OF that[THEN "&E"(1), THEN "&E"(1)]] by fast
6651    }
6652  qed
6653qed
6654
6655AOT_theorem "ind-nec": F ([F]x  [F]y)  F ([F]x  [F]y)
6656proof(rule "→I")
6657  AOT_assume F ([F]x  [F]y)
6658  moreover AOT_have x F ([F]x  [F]y)] by "cqt:2[lambda]"
6659  ultimately AOT_have x F ([F]x  [F]y)]x  x F ([F]x  [F]y)]y
6660    using "∀E" by blast
6661  moreover AOT_have x F ([F]x  [F]y)]y
6662    apply (rule "β←C"(1))
6663      apply "cqt:2[lambda]"
6664     apply (fact "cqt:2[const_var]"[axiom_inst])
6665    by (simp add: RN GEN "oth-class-taut:3:a")
6666  ultimately AOT_have x F ([F]x  [F]y)]x using "≡E" by blast
6667  AOT_thus F ([F]x  [F]y)
6668    using "β→C"(1) by blast
6669qed
6670
6671AOT_theorem "ord=E:1": (O!x & O!y)  (F ([F]x  [F]y)  x =E y)
6672proof (rule "→I"; rule "→I")
6673  AOT_assume F ([F]x  [F]y)
6674  AOT_hence F ([F]x  [F]y)
6675    using "ind-nec"[THEN "→E"] by blast
6676  moreover AOT_assume O!x & O!y
6677  ultimately AOT_have O!x & O!y & F ([F]x  [F]y)
6678    using "&I" by blast
6679  AOT_thus x =E y using "=E-simple:1"[THEN "≡E"(2)] by blast
6680qed
6681
6682AOT_theorem "ord=E:2": (O!x & O!y)  (F ([F]x  [F]y)  x = y)
6683proof (rule "→I"; rule "→I")
6684  AOT_assume O!x & O!y
6685  moreover AOT_assume F ([F]x  [F]y)
6686  ultimately AOT_have x =E y
6687    using "ord=E:1" "→E" by blast
6688  AOT_thus x = y using "=E-simple:2"[THEN "→E"] by blast
6689qed
6690
6691AOT_theorem "ord=E2:1": (O!x & O!y)  (x  y  z z =E x]  z z =E y])
6692proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6693  AOT_assume 0: O!x & O!y
6694  AOT_assume x  y
6695  AOT_hence 1: ¬(x = y) using "≡dfE"[OF "=-infix"] by blast
6696  AOT_assume z z =E x] = z z =E y]
6697  moreover AOT_have z z =E x]x
6698    apply (rule "β←C"(1))
6699      apply "cqt:2[lambda]"
6700     apply (fact "cqt:2[const_var]"[axiom_inst])
6701    using "ord=Eequiv:1"[THEN "→E", OF 0[THEN "&E"(1)]].
6702  ultimately AOT_have z z =E y]x using "rule=E" by fast
6703  AOT_hence x =E y using "β→C"(1) by blast
6704  AOT_hence x = y by (metis "=E-simple:2" "vdash-properties:6")
6705  AOT_thus x = y & ¬(x = y) using 1 "&I" by blast
6706next
6707  AOT_assume z z =E x]  z z =E y]
6708  AOT_hence 0: ¬(z z =E x] = z z =E y]) using "≡dfE"[OF "=-infix"] by blast
6709  AOT_have z z =E x] by "cqt:2[lambda]"
6710  AOT_hence z z =E x] = z z =E x]
6711    by (metis "rule=I:1")
6712  moreover AOT_assume x = y
6713  ultimately AOT_have z z =E x] = z z =E y]
6714    using "rule=E" by fast
6715  AOT_thus z z =E x] = z z =E y] & ¬(z z =E x] = z z =E y])
6716    using 0 "&I" by blast
6717qed
6718
6719AOT_theorem "ord=E2:2": (O!x & O!y)  (x  y  z z = x]  z z = y])
6720proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6721  AOT_assume 0: O!x & O!y
6722  AOT_assume x  y
6723  AOT_hence 1: ¬(x = y) using "≡dfE"[OF "=-infix"] by blast
6724  AOT_assume z z = x] = z z = y]
6725  moreover AOT_have z z = x]x
6726    apply (rule "β←C"(1))
6727    apply (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6728     apply (fact "cqt:2[const_var]"[axiom_inst])
6729    by (simp add: "id-eq:1")
6730  ultimately AOT_have z z = y]x using "rule=E" by fast
6731  AOT_hence x = y using "β→C"(1) by blast
6732  AOT_thus x = y & ¬(x = y) using 1 "&I" by blast
6733next
6734  AOT_assume 0: O!x & O!y
6735  AOT_assume z z = x]  z z = y]
6736  AOT_hence 1: ¬(z z = x] = z z = y]) using "≡dfE"[OF "=-infix"] by blast
6737  AOT_have z z = x] by (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6738  AOT_hence z z = x] = z z = x]
6739    by (metis "rule=I:1")
6740  moreover AOT_assume x = y
6741  ultimately AOT_have z z = x] = z z = y]
6742    using "rule=E" by fast
6743  AOT_thus z z = x] = z z = y] & ¬(z z = x] = z z = y])
6744    using 1 "&I" by blast
6745qed
6746
6747AOT_theorem ordnecfail: O!x  ¬F x[F]
6748  by (meson "RM:1" "deduction-theorem" nocoder "oa-facts:1" "vdash-properties:10" "vdash-properties:1[2]")
6749
6750AOT_theorem "ab-obey:1": (A!x & A!y)  (F (x[F]  y[F])  x = y)
6751proof (rule "→I"; rule "→I")
6752  AOT_assume 1: A!x & A!y
6753  AOT_assume F (x[F]  y[F])
6754  AOT_hence x[F]  y[F] for F using "∀E" by blast
6755  AOT_hence (x[F]  y[F]) for F by (metis "en-eq:6[1]" "≡E"(1))
6756  AOT_hence F (x[F]  y[F]) by (rule GEN)
6757  AOT_hence F (x[F]  y[F]) by (rule BF[THEN "→E"])
6758  AOT_thus x = y
6759    using "≡dfI"[OF "identity:1", OF "∨I"(2)] 1 "&I" by blast
6760qed
6761
6762AOT_theorem "ab-obey:2": (F (x[F] & ¬y[F])  F (y[F] & ¬x[F]))  x  y
6763proof (rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6764  AOT_assume 1: x = y
6765  AOT_assume F (x[F] & ¬y[F])  F (y[F] & ¬x[F])
6766  moreover {
6767    AOT_assume F (x[F] & ¬y[F])
6768    then AOT_obtain F where x[F] & ¬y[F] using "∃E"[rotated] by blast
6769    moreover AOT_have y[F] using calculation[THEN "&E"(1)] 1 "rule=E" by fast
6770    ultimately AOT_have p & ¬p for p by (metis "Conjunction Simplification"(2) "modus-tollens:2" "raa-cor:3")
6771  }
6772  moreover {
6773    AOT_assume F (y[F] & ¬x[F])
6774    then AOT_obtain F where y[F] & ¬x[F] using "∃E"[rotated] by blast
6775    moreover AOT_have ¬y[F] using calculation[THEN "&E"(2)] 1 "rule=E" by fast
6776    ultimately AOT_have p & ¬p for p by (metis "Conjunction Simplification"(1) "modus-tollens:1" "raa-cor:3")
6777  }
6778  ultimately AOT_show p & ¬p for p by (metis "∨E"(3) "raa-cor:1")
6779qed
6780
6781AOT_theorem "encoders-are-abstract": F x[F]  A!x
6782  by (meson "deduction-theorem" "≡E"(2) "modus-tollens:2" nocoder
6783            "oa-contingent:3" "vdash-properties:1[2]")
6784
6785AOT_theorem "denote=:1": Hx x[H]
6786  by (rule GEN; rule "existence:2[1]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6787
6788AOT_theorem "denote=:2": Gx1...∃xn x1...xn[H]
6789  by (rule GEN; rule "existence:2"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6790
6791AOT_theorem "denote=:2[2]": Gx1x2 x1x2[H]
6792  by (rule GEN; rule "existence:2[2]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6793
6794AOT_theorem "denote=:2[3]": Gx1x2x3 x1x2x3[H]
6795  by (rule GEN; rule "existence:2[3]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6796
6797AOT_theorem "denote=:2[4]": Gx1x2x3x4 x1x2x3x4[H]
6798  by (rule GEN; rule "existence:2[4]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6799
6800AOT_theorem "denote=:3": x x[Π]  H (H = Π)
6801  using "existence:2[1]" "free-thms:1" "≡E"(2) "≡E"(5) "Commutativity of ≡" "≡Df" by blast
6802
6803AOT_theorem "denote=:4": (x1...∃xn x1...xn[Π])  H (H = Π)
6804  using "existence:2" "free-thms:1" "≡E"(6) "≡Df" by blast
6805
6806AOT_theorem "denote=:4[2]": (x1x2 x1x2[Π])  H (H = Π)
6807  using "existence:2[2]" "free-thms:1" "≡E"(6) "≡Df" by blast
6808
6809AOT_theorem "denote=:4[3]": (x1x2x3 x1x2x3[Π])  H (H = Π)
6810  using "existence:2[3]" "free-thms:1" "≡E"(6) "≡Df" by blast
6811
6812AOT_theorem "denote=:4[4]": (x1x2x3x4 x1x2x3x4[Π])  H (H = Π)
6813  using "existence:2[4]" "free-thms:1" "≡E"(6) "≡Df" by blast
6814
6815AOT_theorem "A-objects!": ∃!x (A!x & F (x[F]  φ{F}))
6816proof (rule "uniqueness:1"[THEN "≡dfI"])
6817  AOT_obtain a where a_prop: A!a & F (a[F]  φ{F})
6818    using "A-objects"[axiom_inst] "∃E"[rotated] by blast
6819  AOT_have (A!β & F (β[F]  φ{F}))  β = a for β
6820  proof (rule "→I")
6821    AOT_assume β_prop: [A!]β & F (β[F]  φ{F})
6822    AOT_hence β[F]  φ{F} for F using "∀E" "&E" by blast
6823    AOT_hence β[F]  a[F] for F
6824      using a_prop[THEN "&E"(2)] "∀E" "≡E"(2) "≡E"(5) "Commutativity of ≡" by fast
6825    AOT_hence F (β[F]  a[F]) by (rule GEN)
6826    AOT_thus β = a
6827      using "ab-obey:1"[THEN "→E", OF "&I"[OF β_prop[THEN "&E"(1)], OF a_prop[THEN "&E"(1)]], THEN "→E"] by blast
6828  qed
6829  AOT_hence β ((A!β & F (β[F]  φ{F}))  β = a) by (rule GEN)
6830  AOT_thus α ([A!]α & F (α[F]  φ{F}) & β ([A!]β & F (β[F]  φ{F})  β = α))
6831    using "∃I" using a_prop "&I" by fast
6832qed
6833
6834AOT_theorem "obj-oth:1": ∃!x (A!x & F (x[F]  [F]y))
6835  using "A-objects!" by fast
6836
6837AOT_theorem "obj-oth:2": ∃!x (A!x & F (x[F]  [F]y & [F]z))
6838  using "A-objects!" by fast
6839
6840AOT_theorem "obj-oth:3": ∃!x (A!x & F (x[F]  [F]y  [F]z))
6841  using "A-objects!" by fast
6842
6843AOT_theorem "obj-oth:4": ∃!x (A!x & F (x[F]  [F]y))
6844  using "A-objects!" by fast
6845
6846AOT_theorem "obj-oth:5": ∃!x (A!x & F (x[F]  F = G))
6847  using "A-objects!" by fast
6848
6849AOT_theorem "obj-oth:6": ∃!x (A!x & F (x[F]  y([G]y  [F]y)))
6850  using "A-objects!" by fast
6851
6852AOT_theorem "A-descriptions": ιx (A!x & F (x[F]  φ{F}))
6853  by (rule "A-Exists:2"[THEN "≡E"(2)]; rule "RA[2]"; rule "A-objects!")
6854
6855AOT_act_theorem "thm-can-terms2": y = ιx(A!x & F (x[F]  φ{F}))  (A!y & F (y[F]  φ{F}))
6856  using "y-in:2" by blast
6857
6858AOT_theorem "can-ab2": y = ιx(A!x & F (x[F]  φ{F}))   A!y
6859proof(rule "→I")
6860  AOT_assume y = ιx(A!x & F (x[F]  φ{F}))
6861  AOT_hence 𝒜(A!y & F (y[F]  φ{F}))
6862    using "actual-desc:2"[THEN "→E"] by blast
6863  AOT_hence 𝒜A!y by (metis "Act-Basic:2" "&E"(1) "≡E"(1))
6864  AOT_thus A!y by (metis "≡E"(2) "oa-facts:8")
6865qed
6866
6867AOT_act_theorem "desc-encode:1": ιx(A!x & F (x[F]  φ{F}))[F]  φ{F}
6868proof -
6869  AOT_have ιx(A!x & F (x[F]  φ{F}))
6870    by (simp add: "A-descriptions")
6871  AOT_hence A!ιx(A!x & F (x[F]  φ{F})) & F (ιx(A!x & F (x[F]  φ{F}))[F]  φ{F})
6872    using "y-in:3"[THEN "→E"] by blast
6873  AOT_thus ιx(A!x & F (x[F]  φ{F}))[F]  φ{F}
6874    using "&E" "∀E" by blast
6875qed
6876
6877AOT_act_theorem "desc-encode:2": ιx(A!x & F (x[F]  φ{F}))[G]  φ{G}
6878  using "desc-encode:1".
6879
6880AOT_theorem "desc-nec-encode:1": ιx (A!x & F (x[F]  φ{F}))[F]  𝒜φ{F}
6881proof -
6882  AOT_have 0: ιx(A!x & F (x[F]  φ{F}))
6883    by (simp add: "A-descriptions")
6884  AOT_hence 𝒜(A!ιx(A!x & F (x[F]  φ{F})) & F (ιx(A!x & F (x[F]  φ{F}))[F]  φ{F}))
6885    using "actual-desc:4"[THEN "→E"] by blast
6886  AOT_hence 𝒜F (ιx(A!x & F (x[F]  φ{F}))[F]  φ{F})
6887    using "Act-Basic:2" "&E"(2) "≡E"(1) by blast
6888  AOT_hence F 𝒜(ιx(A!x & F (x[F]  φ{F}))[F]  φ{F})
6889    using "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]" by blast
6890  AOT_hence 𝒜(ιx(A!x & F (x[F]  φ{F}))[F]  φ{F})
6891    using "∀E" by blast
6892  AOT_hence 𝒜ιx(A!x & F (x[F]  φ{F}))[F]  𝒜φ{F}
6893    using "Act-Basic:5" "≡E"(1) by blast
6894  AOT_thus ιx(A!x & F (x[F]  φ{F}))[F]  𝒜φ{F}
6895    using "en-eq:10[1]"[unvarify x1, OF 0] "≡E"(6) by blast
6896qed
6897
6898AOT_theorem "desc-nec-encode:2": ιx (A!x & F (x[F]  φ{F}))[G]  𝒜φ{G}
6899  using "desc-nec-encode:1".
6900
6901AOT_theorem "Box-desc-encode:1": φ{G}  ιx(A!x & F (x[F]  φ{G}))[G]
6902  by (rule "→I"; rule "desc-nec-encode:2"[THEN "≡E"(2)])
6903     (meson "nec-imp-act" "vdash-properties:10")
6904
6905AOT_theorem "Box-desc-encode:2": φ{G}  (ιx(A!x & F (x[F]  φ{G}))[G]  φ{G})
6906proof(rule CP)
6907  AOT_assume φ{G}
6908  AOT_hence φ{G} by (metis "S5Basic:6" "≡E"(1))
6909  moreover AOT_have φ{G}  (ιx(A!x & F (x[F]  φ{G}))[G]  φ{G})
6910  proof (rule RM; rule "→I")
6911    AOT_modally_strict {
6912      AOT_assume 1: φ{G}
6913      AOT_hence ιx(A!x & F (x[F]  φ{G}))[G] using "Box-desc-encode:1" "→E" by blast
6914      moreover AOT_have φ{G} using 1 by (meson "qml:2" "vdash-properties:10" "vdash-properties:1[2]")
6915      ultimately AOT_show ιx(A!x & F (x[F]  φ{G}))[G]  φ{G}
6916        using "deduction-theorem" "≡I" by simp
6917    }
6918  qed
6919  ultimately AOT_show (ιx(A!x & F (x[F]  φ{G}))[G]  φ{G}) using "→E" by blast
6920qed
6921
6922definition rigid_condition where rigid_condition φ  v . [v  α (φ{α}  φ{α})]
6923syntax rigid_condition :: ‹id_position  AOT_prop› ("RIGID'_CONDITION'(_')")
6924
6925AOT_theorem "strict-can:1[E]": assumes RIGID_CONDITION(φ)
6926  shows α (φ{α}  φ{α})
6927  using assms[unfolded rigid_condition_def] by auto
6928
6929AOT_theorem "strict-can:1[I]":
6930  assumes  α (φ{α}  φ{α})
6931  shows RIGID_CONDITION(φ)
6932  using assms rigid_condition_def by auto
6933
6934AOT_theorem "box-phi-a:1": assumes RIGID_CONDITION(φ)
6935  shows (A!x  & F (x[F]  φ{F}))  (A!x & F (x[F]  φ{F}))
6936proof (rule "→I")
6937  AOT_assume a: A!x & F (x[F]  φ{F})
6938  AOT_hence b: A!x by (metis "Conjunction Simplification"(1) "oa-facts:2" "vdash-properties:10")
6939  AOT_have x[F]  φ{F} for F using a[THEN "&E"(2)] "∀E" by blast
6940  moreover AOT_have (x[F]  x[F]) for F by (meson "pre-en-eq:1[1]" RN)
6941  moreover AOT_have (φ{F}  φ{F}) for F using RN "strict-can:1[E]"[OF assms] "∀E" by blast
6942  ultimately AOT_have (x[F]  φ{F}) for F
6943    using "sc-eq-box-box:5" "qml:2"[axiom_inst, THEN "→E"] "→E" "&I" by metis
6944  AOT_hence F (x[F]  φ{F}) by (rule GEN)
6945  AOT_hence F (x[F]  φ{F}) by (rule BF[THEN "→E"])
6946  AOT_thus ([A!]x & F (x[F]  φ{F}))
6947    using b "KBasic:3" "≡S"(1) "≡E"(2) by blast
6948qed
6949
6950AOT_theorem "box-phi-a:2": assumes RIGID_CONDITION(φ)
6951  shows y = ιx(A!x & F (x[F]  φ{F}))  (A!y & F (y[F]  φ{F}))
6952proof(rule "→I")
6953  AOT_assume y = ιx(A!x & F (x[F]  φ{F}))
6954  AOT_hence 𝒜(A!y & F (y[F]  φ{F})) using "actual-desc:2"[THEN "→E"] by fast
6955  AOT_hence abs: 𝒜A!y and 𝒜F (y[F]  φ{F})
6956    using "Act-Basic:2" "&E" "≡E"(1) by blast+
6957  AOT_hence F 𝒜(y[F]  φ{F}) by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
6958  AOT_hence 𝒜(y[F]  φ{F}) for F using "∀E" by blast
6959  AOT_hence 𝒜y[F]  𝒜φ{F} for F by (metis "Act-Basic:5" "≡E"(1)) 
6960  AOT_hence y[F]  φ{F} for F
6961    using "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6962    by (metis "en-eq:10[1]" "≡E"(6))
6963  AOT_hence F (y[F]  φ{F}) by (rule GEN)
6964  AOT_thus [A!]y & F (y[F]  φ{F}) using abs "&I" "≡E"(2) "oa-facts:8" by blast
6965qed
6966
6967AOT_theorem "box-phi-a:3": assumes RIGID_CONDITION(φ)
6968  shows ιx(A!x & F (x[F]  φ{F}))[F]  φ{F}
6969  using "desc-nec-encode:2"
6970    "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6971    "≡E"(5) by blast
6972
6973AOT_define Null :: ‹τ  φ› ("Null'(_')") 
6974  "df-null-uni:1": Null(x) df A!x & ¬F x[F]
6975
6976AOT_define Universal :: ‹τ  φ› ("Universal'(_')")
6977  "df-null-uni:2": Universal(x) df A!x & F x[F]
6978
6979AOT_theorem "null-uni-uniq:1": ∃!x Null(x)
6980proof (rule "uniqueness:1"[THEN "≡dfI"])
6981  AOT_obtain a where a_prop: A!a & F (a[F]  ¬(F = F))
6982    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6983  AOT_have a_null: ¬a[F] for F
6984  proof (rule "raa-cor:2")
6985    AOT_assume a[F]
6986    AOT_hence ¬(F = F) using a_prop[THEN "&E"(2)] "∀E" "≡E" by blast
6987    AOT_hence F = F & ¬(F = F) by (metis "id-eq:1" "raa-cor:3")
6988    AOT_thus p & ¬p for p  by (metis "raa-cor:1")
6989  qed
6990  AOT_have Null(a) & β (Null(β)  β = a)
6991  proof (rule "&I")
6992    AOT_have ¬F a[F] using a_null by (metis "instantiation" "reductio-aa:1")
6993    AOT_thus Null(a)
6994      using "df-null-uni:1"[THEN "≡dfI"] a_prop[THEN "&E"(1)] "&I" by metis
6995  next
6996    AOT_show β (Null(β)  β = a)
6997    proof (rule GEN; rule "→I")
6998      fix β
6999      AOT_assume a: Null(β)
7000      AOT_hence ¬F β[F]
7001        using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
7002      AOT_hence β_null: ¬β[F] for F by (metis "existential:2[const_var]" "reductio-aa:1")
7003      AOT_have F (β[F]  a[F])
7004        apply (rule GEN; rule "≡I"; rule CP)
7005        using "raa-cor:3" β_null a_null by blast+
7006      moreover AOT_have A!β using a "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
7007      ultimately AOT_show β = a
7008        using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" by blast
7009    qed
7010  qed
7011  AOT_thus α (Null(α) & β (Null(β)  β = α)) using "∃I"(2) by fast
7012qed
7013
7014AOT_theorem "null-uni-uniq:2": ∃!x Universal(x)
7015proof (rule "uniqueness:1"[THEN "≡dfI"])
7016  AOT_obtain a where a_prop: A!a & F (a[F]  F = F)
7017    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7018  AOT_hence aF: a[F] for F using "&E" "∀E" "≡E" "id-eq:1" by fast
7019  AOT_hence Universal(a)
7020    using "df-null-uni:2"[THEN "≡dfI"] "&I" a_prop[THEN "&E"(1)] GEN by blast
7021  moreover AOT_have β (Universal(β)  β = a)
7022  proof (rule GEN; rule "→I")
7023    fix β
7024    AOT_assume Universal(β)
7025    AOT_hence abs_β: A!β and β[F] for F using "df-null-uni:2"[THEN "≡dfE"] "&E" "∀E" by blast+
7026    AOT_hence β[F]  a[F] for F using aF by (metis "deduction-theorem" "≡I")
7027    AOT_hence F (β[F]  a[F]) by (rule GEN)
7028    AOT_thus β = a
7029      using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" abs_β by blast
7030  qed
7031  ultimately AOT_show α (Universal(α) & β (Universal(β)  β = α))
7032    using "&I" "∃I" by fast
7033qed
7034
7035AOT_theorem "null-uni-uniq:3": ιx Null(x)
7036  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:1" by blast
7037
7038AOT_theorem "null-uni-uniq:4": ιx Universal(x)
7039  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:2" by blast
7040
7041AOT_define Null_object :: ‹κs (a)
7042  "df-null-uni-terms:1": a =df ιx Null(x)
7043
7044AOT_define Universal_object :: ‹κs (aV)
7045  "df-null-uni-terms:2": aV =df ιx Universal(x)
7046
7047AOT_theorem "null-uni-facts:1": Null(x)  Null(x)
7048proof (rule "→I")
7049  AOT_assume Null(x)
7050  AOT_hence x_abs: A!x and x_null: ¬F x[F]
7051    using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast+
7052  AOT_have ¬x[F] for F using x_null
7053    using "existential:2[const_var]" "reductio-aa:1"
7054    by metis
7055  AOT_hence ¬x[F] for F by (metis "en-eq:7[1]" "≡E"(1))
7056  AOT_hence F ¬x[F] by (rule GEN)
7057  AOT_hence F ¬x[F] by (rule BF[THEN "→E"])
7058  moreover AOT_have F ¬x[F]  ¬F x[F]
7059    apply (rule RM)
7060    by (metis (full_types) "instantiation" "cqt:2[const_var]" "deduction-theorem"
7061                           "reductio-aa:1" "rule-ui:1" "vdash-properties:1[2]")
7062  ultimately AOT_have ¬F x[F]
7063    by (metis "→E")
7064  moreover AOT_have A!x using x_abs
7065    using "oa-facts:2" "vdash-properties:10" by blast
7066  ultimately AOT_have r: (A!x & ¬F x[F])
7067    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
7068  AOT_show Null(x)
7069    by (AOT_subst Null(x) A!x & ¬F x[F])
7070       (auto simp: "df-null-uni:1" "≡Df" r)
7071qed  
7072
7073AOT_theorem "null-uni-facts:2": Universal(x)  Universal(x)
7074proof (rule "→I")
7075  AOT_assume Universal(x)
7076  AOT_hence x_abs: A!x and x_univ: F x[F]
7077    using "df-null-uni:2"[THEN "≡dfE"] "&E" by blast+
7078  AOT_have x[F] for F using x_univ "∀E" by blast
7079  AOT_hence x[F] for F by (metis "en-eq:2[1]" "≡E"(1))
7080  AOT_hence F x[F] by (rule GEN)
7081  AOT_hence F x[F] by (rule BF[THEN "→E"])
7082  moreover AOT_have A!x using x_abs
7083    using "oa-facts:2" "vdash-properties:10" by blast
7084  ultimately AOT_have r: (A!x & F x[F])
7085    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
7086  AOT_show Universal(x)
7087    by (AOT_subst Universal(x) A!x & F x[F])
7088       (auto simp add: "df-null-uni:2" "≡Df" r)
7089qed
7090
7091AOT_theorem "null-uni-facts:3": Null(a)
7092  apply (rule "=dfI"(2)[OF "df-null-uni-terms:1"])
7093   apply (simp add: "null-uni-uniq:3")
7094  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:3"]
7095    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:1"[unvarify x, THEN RN, OF "null-uni-uniq:3"], THEN "≡E"(1)]
7096  by blast
7097
7098AOT_theorem "null-uni-facts:4": Universal(aV)
7099  apply (rule "=dfI"(2)[OF "df-null-uni-terms:2"])
7100   apply (simp add: "null-uni-uniq:4")
7101  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:4"]
7102    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:2"[unvarify x, THEN RN, OF "null-uni-uniq:4"], THEN "≡E"(1)]
7103  by blast
7104
7105AOT_theorem "null-uni-facts:5": a  aV
7106proof (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
7107    rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
7108    rule "≡dfI"[OF "=-infix"];
7109    rule "raa-cor:2")
7110  AOT_obtain x where nullx: Null(x)
7111    by (metis "instantiation" "df-null-uni-terms:1" "existential:1" "null-uni-facts:3"
7112              "null-uni-uniq:3" "rule-id-df:2:b[zero]")
7113  AOT_hence act_null: 𝒜Null(x) by (metis "nec-imp-act" "null-uni-facts:1" "vdash-properties:10")
7114  AOT_assume ιx Null(x) = ιx Universal(x)
7115  AOT_hence 𝒜x(Null(x)  Universal(x))
7116    using "actual-desc:5"[THEN "→E"] by blast
7117  AOT_hence x 𝒜(Null(x)  Universal(x))
7118    by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
7119  AOT_hence 𝒜Null(x)  𝒜Universal(x)
7120    using "Act-Basic:5" "≡E"(1) "rule-ui:3" by blast
7121  AOT_hence 𝒜Universal(x) using act_null "≡E" by blast
7122  AOT_hence Universal(x) by (metis RN "≡E"(1) "null-uni-facts:2" "sc-eq-fur:2" "vdash-properties:10")
7123  AOT_hence F x[F] using "≡dfE"[OF "df-null-uni:2"] "&E" by metis
7124  moreover AOT_have ¬F x[F] using nullx "≡dfE"[OF "df-null-uni:1"] "&E" by metis
7125  ultimately AOT_show p & ¬p for p by (metis "cqt-further:1" "raa-cor:3" "vdash-properties:10")
7126qed
7127
7128AOT_theorem "null-uni-facts:6": a = ιx(A!x & F (x[F]  F  F))
7129proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7130  AOT_show ιx([A!]x & F (x[F]  F  F))
7131    by (simp add: "A-descriptions")
7132next
7133  AOT_show a
7134    by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7135       (simp add: "null-uni-uniq:3")
7136next
7137  AOT_have ιx([A!]x & F (x[F]  F  F))
7138    by (simp add: "A-descriptions")
7139  AOT_hence 1: ιx([A!]x & F (x[F]  F  F)) = ιx([A!]x & F (x[F]  F  F))
7140    using "rule=I:1" by blast
7141  AOT_show [A!]a & [A!]ιx([A!]x & F (x[F]  F  F))
7142    apply (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"]; rule "&I")
7143    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3" "null-uni-uniq:3" "rule-id-df:2:a[zero]" "vdash-properties:10")
7144    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7145next
7146  AOT_show F (a[F]  ιx([A!]x & F (x[F]  F  F))[F])
7147  proof (rule GEN)
7148    fix F
7149    AOT_have ¬a[F]
7150      by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7151         (metis (no_types, lifting) "≡dfE" "&E"(2) "∨I"(2) "∨E"(3)
7152                "df-null-uni:1" "df-null-uni-terms:1" "existential:2[const_var]" "null-uni-facts:3"
7153                "raa-cor:2" "rule-id-df:2:a[zero]" "russell-axiom[enc,1].ψ_denotes_asm")
7154    moreover AOT_have ¬ιx([A!]x & F (x[F]  F  F))[F]
7155    proof(rule "raa-cor:2")
7156      AOT_assume 0: ιx([A!]x & F (x[F]  F  F))[F]
7157      AOT_hence 𝒜(F  F) using "desc-nec-encode:2"[THEN "≡E"(1), OF 0] by blast
7158      moreover AOT_have ¬𝒜(F  F)
7159        using "≡dfE" "id-act:2" "id-eq:1" "≡E"(2) "=-infix" "raa-cor:3" by blast
7160      ultimately AOT_show 𝒜(F  F) & ¬𝒜(F  F) by (rule "&I")
7161    qed
7162    ultimately AOT_show a[F]  ιx([A!]x & F (x[F]  F  F))[F]
7163      using "deduction-theorem" "≡I" "raa-cor:4" by blast
7164  qed
7165qed
7166
7167AOT_theorem "null-uni-facts:7": aV = ιx(A!x & F (x[F]  F = F))
7168proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7169  AOT_show ιx([A!]x & F (x[F]  F = F))
7170    by (simp add: "A-descriptions")
7171next
7172  AOT_show aV
7173    by (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7174       (simp add: "null-uni-uniq:4")
7175next
7176  AOT_have ιx([A!]x & F (x[F]  F = F))
7177    by (simp add: "A-descriptions")
7178  AOT_hence 1: ιx([A!]x & F (x[F]  F = F)) = ιx([A!]x & F (x[F]  F = F))
7179    using "rule=I:1" by blast
7180  AOT_show [A!]aV & [A!]ιx([A!]x & F (x[F]  F = F))
7181    apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"]; rule "&I")
7182    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "vdash-properties:10")
7183    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7184next
7185  AOT_show F (aV[F]  ιx([A!]x & F (x[F]  F = F))[F])
7186  proof (rule GEN)
7187    fix F
7188    AOT_have aV[F]
7189      apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7190      using "≡dfE" "&E"(2) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "rule-ui:3" by blast
7191    moreover AOT_have ιx([A!]x & F (x[F]  F = F))[F]
7192      using "RA[2]" "desc-nec-encode:2" "id-eq:1" "≡E"(2) by fastforce
7193    ultimately AOT_show aV[F]  ιx([A!]x & F (x[F]  F = F))[F]
7194      using "deduction-theorem" "≡I" by simp
7195  qed
7196qed
7197
7198AOT_theorem "aclassical:1": Rxy(A!x & A!y & x  y & z [R]zx] = z [R]zy])
7199proof(rule GEN)
7200  fix R
7201  AOT_obtain a where a_prop: A!a & F (a[F]  y(A!y & F = z [R]zy] & ¬y[F]))
7202    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7203  AOT_have a_enc: az [R]za]
7204  proof (rule "raa-cor:1")
7205    AOT_assume 0: ¬az [R]za]
7206    AOT_hence ¬y(A!y & z [R]za] = z [R]zy] & ¬yz [R]za])
7207      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«z [R]za]»"],
7208                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7209         "cqt:2[lambda]"
7210    AOT_hence y ¬(A!y & z [R]za] = z [R]zy] & ¬yz [R]za])
7211      using "cqt-further:4" "vdash-properties:10" by blast
7212    AOT_hence ¬(A!a & z [R]za] = z [R]za] & ¬az [R]za]) using "∀E" by blast
7213    AOT_hence (A!a & z [R]za] = z [R]za])  az [R]za]
7214      by (metis "&I" "deduction-theorem" "raa-cor:3")
7215    moreover AOT_have z [R]za] = z [R]za]
7216      by (rule "=I") "cqt:2[lambda]"
7217    ultimately AOT_have az [R]za] using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7218    AOT_thus az [R]za] & ¬az [R]za]
7219      using 0 "&I" by blast
7220  qed
7221  AOT_hence y(A!y & z [R]za] = z [R]zy] & ¬yz [R]za])
7222    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7223  then AOT_obtain b where b_prop: A!b & z [R]za] = z [R]zb] & ¬bz [R]za]
7224    using "∃E"[rotated] by blast
7225  AOT_have a  b
7226    apply (rule "≡dfI"[OF "=-infix"])
7227    using a_enc b_prop[THEN "&E"(2)]
7228    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7229  AOT_hence A!a & A!b & a  b & z [R]za] = z [R]zb]
7230    using b_prop "&E" a_prop "&I" by meson
7231  AOT_hence y (A!a & A!y & a  y & z [R]za] = z [R]zy]) by (rule "∃I")
7232  AOT_thus xy (A!x & A!y & x  y & z [R]zx] = z [R]zy]) by (rule "∃I")
7233qed
7234
7235AOT_theorem "aclassical:2": Rxy(A!x & A!y & x  y & z [R]xz] = z [R]yz])
7236proof(rule GEN)
7237  fix R
7238  AOT_obtain a where a_prop: A!a & F (a[F]  y(A!y & F = z [R]yz] & ¬y[F]))
7239    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7240  AOT_have a_enc: az [R]az]
7241  proof (rule "raa-cor:1")
7242    AOT_assume 0: ¬az [R]az]
7243    AOT_hence ¬y(A!y & z [R]az] = z [R]yz] & ¬yz [R]az])
7244      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«z [R]az]»"],
7245                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7246         "cqt:2[lambda]"
7247    AOT_hence y ¬(A!y & z [R]az] = z [R]yz] & ¬yz [R]az])
7248      using "cqt-further:4" "vdash-properties:10" by blast
7249    AOT_hence ¬(A!a & z [R]az] = z [R]az] & ¬az [R]az]) using "∀E" by blast
7250    AOT_hence (A!a & z [R]az] = z [R]az])  az [R]az]
7251      by (metis "&I" "deduction-theorem" "raa-cor:3")
7252    moreover AOT_have z [R]az] = z [R]az]
7253      by (rule "=I") "cqt:2[lambda]"
7254    ultimately AOT_have az [R]az] using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7255    AOT_thus az [R]az] & ¬az [R]az]
7256      using 0 "&I" by blast
7257  qed
7258  AOT_hence y(A!y & z [R]az] = z [R]yz] & ¬yz [R]az])
7259    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7260  then AOT_obtain b where b_prop: A!b & z [R]az] = z [R]bz] & ¬bz [R]az]
7261    using "∃E"[rotated] by blast
7262  AOT_have a  b
7263    apply (rule "≡dfI"[OF "=-infix"])
7264    using a_enc b_prop[THEN "&E"(2)]
7265    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7266  AOT_hence A!a & A!b & a  b & z [R]az] = z [R]bz]
7267    using b_prop "&E" a_prop "&I" by meson
7268  AOT_hence y (A!a & A!y & a  y & z [R]az] = z [R]yz]) by (rule "∃I")
7269  AOT_thus xy (A!x & A!y & x  y & z [R]xz] = z [R]yz]) by (rule "∃I")
7270qed
7271
7272AOT_theorem "aclassical:3": Fxy(A!x & A!y & x  y &  [F]x] =  [F]y])
7273proof(rule GEN)
7274  fix R
7275  AOT_obtain a where a_prop: A!a & F (a[F]  y(A!y & F = z [R]y] & ¬y[F]))
7276    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7277  AOT_have z [R]a] by "cqt:2[lambda]"
7278  (* TODO: S should no longer be necessary *)
7279  then AOT_obtain S where S_def: S = z [R]a]
7280    by (metis "instantiation" "rule=I:1" "existential:1" id_sym)
7281  AOT_have a_enc: a[S]
7282  proof (rule "raa-cor:1")
7283    AOT_assume 0: ¬a[S]
7284    AOT_hence ¬y(A!y & S = z [R]y] & ¬y[S])
7285      by (rule a_prop[THEN "&E"(2), THEN "∀E"(2)[where β=S],
7286                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated]) 
7287    AOT_hence y ¬(A!y & S = z [R]y] & ¬y[S])
7288      using "cqt-further:4" "vdash-properties:10" by blast
7289    AOT_hence ¬(A!a & S = z [R]a] & ¬a[S]) using "∀E" by blast
7290    AOT_hence (A!a & S = z [R]a])  a[S]
7291      by (metis "&I" "deduction-theorem" "raa-cor:3")
7292    moreover AOT_have S = z [R]a] using S_def .
7293    ultimately AOT_have a[S] using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7294    AOT_thus az [R]a] & ¬az [R]a]  by (metis "0" "raa-cor:3") 
7295  qed
7296  AOT_hence y(A!y & S = z [R]y] & ¬y[S])
7297    by (rule a_prop[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1), rotated])
7298  then AOT_obtain b where b_prop: A!b & S = z [R]b] & ¬b[S]
7299    using "∃E"[rotated] by blast
7300  AOT_have 1: a  b
7301    apply (rule "≡dfI"[OF "=-infix"])
7302    using a_enc b_prop[THEN "&E"(2)]
7303    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7304  AOT_have a:  [R]a] = ([R]a)
7305    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7306    by (meson "log-prop-prop:2")
7307  AOT_have b:  [R]b] = ([R]b)
7308    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7309    by (meson "log-prop-prop:2")
7310  AOT_have  [R]a] =  [R]b]
7311    apply (rule "rule=E"[rotated, OF a[THEN id_sym]])
7312    apply (rule "rule=E"[rotated, OF b[THEN id_sym]])
7313    apply (rule "identity:4"[THEN "≡dfI", OF "&I", rotated])
7314     apply (rule "rule=E"[rotated, OF S_def])
7315    using b_prop "&E" apply blast
7316    apply (safe intro!: "&I")
7317    by (simp add: "log-prop-prop:2")+
7318  AOT_hence A!a & A!b & a  b &  [R]a] =  [R]b]
7319    using 1 a_prop[THEN "&E"(1)] b_prop[THEN "&E"(1), THEN "&E"(1)] "&I" by auto
7320  AOT_hence y (A!a & A!y & a  y &  [R]a] =  [R]y]) by (rule "∃I")
7321  AOT_thus xy (A!x & A!y & x  y &  [R]x] =  [R]y]) by (rule "∃I")
7322qed
7323
7324AOT_theorem aclassical2: xy (A!x & A!y & x  y & F ([F]x  [F]y))
7325proof -
7326  AOT_have x y ([A!]x & [A!]y & x  y &
7327               z xy F ([F]x  [F]y)]zx] = z xy F ([F]x  [F]y)]zy])
7328    by (rule "aclassical:1"[THEN "∀E"(1)[where τ="«xy F ([F]x  [F]y)]»"]])
7329       "cqt:2[lambda]"
7330  then AOT_obtain x where y ([A!]x & [A!]y & x  y &
7331               z xy F ([F]x  [F]y)]zx] = z xy F ([F]x  [F]y)]zy])
7332    using "∃E"[rotated] by blast
7333  then AOT_obtain y where 0: ([A!]x & [A!]y & x  y &
7334               z xy F ([F]x  [F]y)]zx] = z xy F ([F]x  [F]y)]zy])
7335    using "∃E"[rotated] by blast
7336  AOT_have z xy F ([F]x  [F]y)]zx]x
7337    apply (rule "β←C"(1))
7338      apply "cqt:2[lambda]"
7339     apply (fact "cqt:2[const_var]"[axiom_inst])
7340    apply (rule "β←C"(1))
7341      apply "cqt:2[lambda]"
7342    apply (simp add: "&I" "ex:1:a" prod_denotesI "rule-ui:3")
7343    by (simp add: "oth-class-taut:3:a" "universal-cor")
7344  AOT_hence z xy F ([F]x  [F]y)]zy]x
7345    by (rule "rule=E"[rotated, OF 0[THEN "&E"(2)]])
7346  AOT_hence xy F ([F]x  [F]y)]xy
7347    by (rule "β→C"(1))
7348  AOT_hence F ([F]x  [F]y)
7349    using "β→C"(1) old.prod.case by fast
7350  AOT_hence [A!]x & [A!]y & x  y & F ([F]x  [F]y) using 0 "&E" "&I" by blast
7351  AOT_hence y ([A!]x & [A!]y & x  y & F ([F]x  [F]y)) by (rule "∃I")
7352  AOT_thus xy ([A!]x & [A!]y & x  y & F ([F]x  [F]y)) by (rule "∃I"(2))
7353qed
7354
7355AOT_theorem "kirchner-thm:1": x φ{x}]  xy(F([F]x  [F]y)  (φ{x}  φ{y}))
7356proof(rule "≡I"; rule "→I")
7357  AOT_assume x φ{x}]
7358  AOT_hence x φ{x}] by (metis "exist-nec" "vdash-properties:10")
7359  moreover AOT_have x φ{x}]  xy(F([F]x  [F]y)  (φ{x}  φ{y}))
7360  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7361    AOT_modally_strict {
7362      fix x y
7363      AOT_assume 0: x φ{x}]
7364      moreover AOT_assume F([F]x  [F]y)
7365      ultimately AOT_have x φ{x}]x  x φ{x}]y
7366        using "∀E" by blast
7367      AOT_thus (φ{x}  φ{y})
7368        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7369    }
7370  qed
7371  ultimately AOT_show xy(F([F]x  [F]y)  (φ{x}  φ{y}))
7372    using "→E" by blast
7373next
7374  AOT_have xy(F([F]x  [F]y)  (φ{x}  φ{y}))  y(x(F([F]x  [F]y) & φ{x})  φ{y})
7375  proof(rule "RM:1"; rule "→I"; rule GEN)
7376    AOT_modally_strict {
7377      AOT_assume xy(F([F]x  [F]y)  (φ{x}  φ{y}))
7378      AOT_hence indisc: φ{x}  φ{y} if F([F]x  [F]y) for x y
7379        using "∀E"(2) "→E" that by blast
7380      AOT_show (x(F([F]x  [F]y) & φ{x})  φ{y}) for y
7381      proof (rule "raa-cor:1")
7382        AOT_assume ¬(x(F([F]x  [F]y) & φ{x})  φ{y})
7383        AOT_hence (x(F([F]x  [F]y) & φ{x}) & ¬φ{y})  (¬(x(F([F]x  [F]y) & φ{x})) & φ{y})
7384          using "≡E"(1) "oth-class-taut:4:h" by blast
7385        moreover {
7386          AOT_assume 0: x(F([F]x  [F]y) & φ{x}) & ¬φ{y}
7387          AOT_obtain a where F([F]a  [F]y) & φ{a}
7388            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7389          AOT_hence φ{y} using indisc[THEN "≡E"(1)] "&E" by blast
7390          AOT_hence p & ¬p for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7391        }
7392        moreover {
7393          AOT_assume 0: (¬(x(F([F]x  [F]y) & φ{x})) & φ{y})
7394          AOT_hence x ¬(F([F]x  [F]y) & φ{x})
7395            using "&E"(1) "cqt-further:4" "→E" by blast
7396          AOT_hence ¬(F([F]y  [F]y) & φ{y}) using "∀E" by blast
7397          AOT_hence ¬F([F]y  [F]y)  ¬φ{y}
7398            using "≡E"(1) "oth-class-taut:5:c" by blast
7399          moreover AOT_have F([F]y  [F]y) by (simp add: "oth-class-taut:3:a" "universal-cor")
7400          ultimately AOT_have ¬φ{y} by (metis "¬¬I" "∨E"(2))
7401          AOT_hence p & ¬p for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7402        }
7403        ultimately AOT_show p & ¬p for p using "∨E"(3) "raa-cor:1" by blast
7404      qed
7405    }
7406  qed
7407  moreover AOT_assume xy(F([F]x  [F]y)  (φ{x}  φ{y}))
7408  ultimately AOT_have y(x(F([F]x  [F]y) & φ{x})  φ{y})
7409    using "→E" by blast
7410  AOT_thus x φ{x}]
7411    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7412qed
7413
7414AOT_theorem "kirchner-thm:2": x1...xn φ{x1...xn}]  x1...∀xny1...∀yn(F([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7415proof(rule "≡I"; rule "→I")
7416  AOT_assume x1...xn φ{x1...xn}]
7417  AOT_hence x1...xn φ{x1...xn}] by (metis "exist-nec" "vdash-properties:10")
7418  moreover AOT_have x1...xn φ{x1...xn}]  x1...∀xny1...∀yn(F([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7419  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7420    AOT_modally_strict {
7421      fix x1xn y1yn :: 'a AOT_var›
7422      AOT_assume 0: x1...xn φ{x1...xn}]
7423      moreover AOT_assume F([F]x1...xn  [F]y1...yn)
7424      ultimately AOT_have x1...xn φ{x1...xn}]x1...xn  x1...xn φ{x1...xn}]y1...yn
7425        using "∀E" by blast
7426      AOT_thus (φ{x1...xn}  φ{y1...yn})
7427        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7428    }
7429  qed
7430  ultimately AOT_show x1...∀xny1...∀yn(F([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7431    using "→E" by blast
7432next
7433  AOT_have (x1...∀xny1...∀yn(F([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))) 
7434            y1...∀yn((x1...∃xn(F([F]x1...xn  [F]y1...yn) & φ{x1...xn}))  φ{y1...yn})
7435  proof(rule "RM:1"; rule "→I"; rule GEN)
7436    AOT_modally_strict {
7437      AOT_assume x1...∀xny1...∀yn(F([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7438      AOT_hence indisc: φ{x1...xn}  φ{y1...yn} if F([F]x1...xn  [F]y1...yn) for x1xn y1yn
7439        using "∀E"(2) "→E" that by blast
7440      AOT_show (x1...∃xn(F([F]x1...xn  [F]y1...yn) & φ{x1...xn}))  φ{y1...yn} for y1yn
7441      proof (rule "raa-cor:1")
7442        AOT_assume ¬((x1...∃xn(F([F]x1...xn  [F]y1...yn) & φ{x1...xn}))  φ{y1...yn})
7443        AOT_hence ((x1...∃xn(F([F]x1...xn  [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}) 
7444                    (¬(x1...∃xn(F([F]x1...xn  [F]y1...yn) & φ{x1...xn})) & φ{y1...yn})
7445          using "≡E"(1) "oth-class-taut:4:h" by blast
7446        moreover {
7447          AOT_assume 0: (x1...∃xn(F([F]x1...xn  [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}
7448          AOT_obtain a1an where F([F]a1...an  [F]y1...yn) & φ{a1...an}
7449            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7450          AOT_hence φ{y1...yn} using indisc[THEN "≡E"(1)] "&E" by blast
7451          AOT_hence p & ¬p for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7452        }
7453        moreover {
7454          AOT_assume 0: (¬((x1...∃xn(F([F]x1...xn  [F]y1...yn) & φ{x1...xn}))) & φ{y1...yn})
7455          AOT_hence x1...∀xn ¬(F([F]x1...xn  [F]y1...yn) & φ{x1...xn})
7456            using "&E"(1) "cqt-further:4" "→E" by blast
7457          AOT_hence ¬(F([F]y1...yn  [F]y1...yn) & φ{y1...yn}) using "∀E" by blast
7458          AOT_hence ¬F([F]y1...yn  [F]y1...yn)  ¬φ{y1...yn}
7459            using "≡E"(1) "oth-class-taut:5:c" by blast
7460          moreover AOT_have F([F]y1...yn  [F]y1...yn)
7461            by (simp add: "oth-class-taut:3:a" "universal-cor")
7462          ultimately AOT_have ¬φ{y1...yn} by (metis "¬¬I" "∨E"(2))
7463          AOT_hence p & ¬p for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7464        }
7465        ultimately AOT_show p & ¬p for p using "∨E"(3) "raa-cor:1" by blast
7466      qed
7467    }
7468  qed
7469  moreover AOT_assume x1...∀xny1...∀yn(F([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7470  ultimately AOT_have y1...∀yn((x1...∃xn(F([F]x1...xn  [F]y1...yn) & φ{x1...xn}))  φ{y1...yn})
7471    using "→E" by blast
7472  AOT_thus x1...xn φ{x1...xn}]
7473    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7474qed
7475
7476AOT_theorem "kirchner-thm-cor:1": x φ{x}]  xy(F([F]x  [F]y)  (φ{x}  φ{y}))
7477proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7478  fix x y
7479  AOT_assume x φ{x}]
7480  AOT_hence xy (F ([F]x  [F]y)  (φ{x}  φ{y}))
7481    by (rule "kirchner-thm:1"[THEN "≡E"(1)])
7482  AOT_hence xy (F ([F]x  [F]y)  (φ{x}  φ{y}))
7483    using CBF[THEN "→E"] by blast
7484  AOT_hence y (F ([F]x  [F]y)  (φ{x}  φ{y}))
7485    using "∀E" by blast
7486  AOT_hence y (F ([F]x  [F]y)  (φ{x}  φ{y}))
7487    using CBF[THEN "→E"] by blast
7488  AOT_hence (F ([F]x  [F]y)  (φ{x}  φ{y}))
7489    using "∀E" by blast
7490  AOT_hence F ([F]x  [F]y)  (φ{x}  φ{y})
7491    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7492  moreover AOT_assume F([F]x  [F]y)
7493  ultimately AOT_show (φ{x}  φ{y}) using "→E" "ind-nec" by blast
7494qed
7495
7496AOT_theorem "kirchner-thm-cor:2":
7497  x1...xn φ{x1...xn}]  x1...∀xny1...∀yn(F([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7498proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7499  fix x1xn y1yn
7500  AOT_assume x1...xn φ{x1...xn}]
7501  AOT_hence 0: x1...∀xny1...∀yn (F ([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7502    by (rule "kirchner-thm:2"[THEN "≡E"(1)])
7503  AOT_have x1...∀xny1...∀yn (F ([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7504  proof(rule GEN; rule GEN)
7505    fix x1xn y1yn
7506    AOT_show (F ([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7507      apply (rule "RM:1"[THEN "→E", rotated, OF 0]; rule "→I")
7508      using "∀E" by blast
7509  qed
7510  AOT_hence y1...∀yn (F ([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7511    using "∀E" by blast
7512  AOT_hence (F ([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7513    using "∀E" by blast
7514  AOT_hence (F ([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn}))
7515    using "∀E" by blast
7516  AOT_hence 0: F ([F]x1...xn  [F]y1...yn)  (φ{x1...xn}  φ{y1...yn})
7517    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7518  moreover AOT_assume F([F]x1...xn  [F]y1...yn)
7519  moreover AOT_have x1...xn F ([F]x1...xn  [F]y1...yn)] by "cqt:2[lambda]"
7520  ultimately AOT_have x1...xn F ([F]x1...xn  [F]y1...yn)]x1...xn  x1...xn F ([F]x1...xn  [F]y1...yn)]y1...yn
7521    using "∀E" by blast
7522  moreover AOT_have x1...xn F ([F]x1...xn  [F]y1...yn)]y1...yn
7523    apply (rule "β←C"(1))
7524      apply "cqt:2[lambda]"
7525     apply (fact "cqt:2[const_var]"[axiom_inst])
7526    by (simp add: RN GEN "oth-class-taut:3:a")
7527  ultimately AOT_have x1...xn F ([F]x1...xn  [F]y1...yn)]x1...xn using "≡E"(2) by blast
7528  AOT_hence F ([F]x1...xn  [F]y1...yn)
7529    using "β→C"(1) by blast
7530  AOT_thus (φ{x1...xn}  φ{y1...yn}) using "→E" 0 by blast
7531qed
7532
7533AOT_define propositional :: ‹Π  φ› (Propositional'(_'))
7534  "prop-prop1": Propositional([F]) df p(F = y p])
7535
7536AOT_theorem "prop-prop2:1": p y p]
7537  by (rule GEN) "cqt:2[lambda]"
7538
7539AOT_theorem "prop-prop2:2": ν φ]
7540  by "cqt:2[lambda]"
7541
7542AOT_theorem "prop-prop2:3": F = y p]  x([F]x  p)
7543proof (rule "→I")
7544  AOT_assume 0: F = y p]
7545  AOT_show x([F]x  p)
7546    by (rule "rule=E"[rotated, OF 0[symmetric]]; rule RN; rule GEN; rule "beta-C-meta"[THEN "→E"])
7547      "cqt:2[lambda]"
7548qed
7549
7550AOT_theorem "prop-prop2:4": Propositional([F])  Propositional([F])
7551proof(rule "→I")
7552  AOT_assume Propositional([F])
7553  AOT_hence p(F = y p]) using "≡dfE"[OF "prop-prop1"] by blast
7554  then AOT_obtain p where F = y p] using "∃E"[rotated] by blast
7555  AOT_hence (F = y p]) using "id-nec:2" "modus-tollens:1" "raa-cor:3" by blast
7556  AOT_hence p (F = y p]) using "∃I" by fast
7557  AOT_hence 0: p (F = y p]) by (metis Buridan "vdash-properties:10")
7558  AOT_thus Propositional([F])
7559    using "prop-prop1"[THEN "≡Df"]
7560    by (AOT_subst Propositional([F]) p (F = y p])) auto
7561qed
7562
7563AOT_define indicriminate :: ‹Π  φ› ("Indiscriminate'(_')")
7564  "prop-indis": Indiscriminate([F]) df F & (x [F]x  x [F]x)
7565
7566AOT_theorem "prop-in-thm": Propositional([Π])  Indiscriminate([Π])
7567proof(rule "→I")
7568  AOT_assume Propositional([Π])
7569  AOT_hence p Π = y p] using "≡dfE"[OF "prop-prop1"] by blast
7570  then AOT_obtain p where Π_def: Π = y p] using "∃E"[rotated] by blast
7571  AOT_show Indiscriminate([Π])
7572  proof (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7573    AOT_show Π
7574      using Π_def by (meson "t=t-proper:1" "vdash-properties:6")
7575  next
7576    AOT_show (x [Π]x  x [Π]x)
7577    proof (rule "rule=E"[rotated, OF Π_def[symmetric]]; rule RN; rule "→I"; rule GEN)
7578      AOT_modally_strict {
7579        AOT_assume x y p]x
7580        then AOT_obtain a where y p]a using "∃E"[rotated] by blast
7581        AOT_hence 0: p by (metis "β→C"(1))
7582        AOT_show y p]x for x
7583          apply (rule "β←C"(1))
7584            apply "cqt:2[lambda]"
7585           apply (fact "cqt:2[const_var]"[axiom_inst])
7586          by (fact 0)
7587      }
7588    qed
7589  qed
7590qed
7591
7592AOT_theorem "prop-in-f:1": Necessary([F])  Indiscriminate([F])
7593proof (rule "→I")
7594  AOT_assume Necessary([F])
7595  AOT_hence 0: x1...∀xn [F]x1...xn using "≡dfE"[OF "contingent-properties:1"] by blast
7596  AOT_show Indiscriminate([F])
7597    by (rule "≡dfI"[OF "prop-indis"])
7598       (metis "0" "KBasic:1" "&I" "ex:1:a" "rule-ui:2[const_var]" "vdash-properties:6") 
7599qed
7600
7601AOT_theorem "prop-in-f:2": Impossible([F])  Indiscriminate([F])
7602proof (rule "→I")
7603  AOT_modally_strict {
7604    AOT_have x ¬[F]x  (x [F]x  x [F]x)
7605      by (metis "instantiation" "cqt-orig:3" "Hypothetical Syllogism" "deduction-theorem" "raa-cor:3")
7606  }
7607  AOT_hence 0: x ¬[F]x  (x [F]x  x [F]x)
7608    by (rule "RM:1")
7609  AOT_assume Impossible([F])
7610  AOT_hence x ¬[F]x using "≡dfE"[OF "contingent-properties:2"] "&E" by blast
7611  AOT_hence 1: (x [F]x  x [F]x) using 0 "→E" by blast
7612  AOT_show Indiscriminate([F])
7613    by (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7614       (simp add: "ex:1:a" "rule-ui:2[const_var]" 1)+
7615qed
7616
7617AOT_theorem "prop-in-f:3:a": ¬Indiscriminate([E!])
7618proof(rule "raa-cor:2")
7619  AOT_assume Indiscriminate([E!])
7620  AOT_hence 0: (x [E!]x  x [E!]x)
7621    using "≡dfE"[OF "prop-indis"] "&E" by blast
7622  AOT_hence x [E!]x  x [E!]x
7623    using "KBasic:13" "vdash-properties:10" by blast
7624  moreover AOT_have x [E!]x
7625    by (simp add: "thm-cont-e:3")
7626  ultimately AOT_have x [E!]x
7627    by (metis "vdash-properties:6")
7628  AOT_thus p & ¬p for p
7629    by (metis "≡dfE" "conventions:5" "o-objects-exist:5" "reductio-aa:1")
7630qed
7631
7632AOT_theorem "prop-in-f:3:b": ¬Indiscriminate([E!]-)
7633proof (rule "rule=E"[rotated, OF "rel-neg-T:2"[symmetric]]; rule "raa-cor:2")
7634  AOT_assume Indiscriminate(x ¬[E!]x])
7635  AOT_hence 0: (x x ¬[E!]x]x  x x ¬[E!]x]x)
7636    using "≡dfE"[OF "prop-indis"] "&E" by blast
7637  AOT_hence x x ¬[E!]x]x  x x ¬[E!]x]x
7638    using "→E" "qml:1" "vdash-properties:1[2]" by blast
7639  moreover AOT_have x x ¬[E!]x]x
7640    apply (AOT_subst x ¬E!x]x ¬E!x bound: x)
7641    apply (rule "beta-C-meta"[THEN "→E"])
7642     apply "cqt:2[lambda]"
7643    by (metis (full_types) "B◇" RN "T◇" "cqt-further:2" "o-objects-exist:5" "vdash-properties:10")
7644  ultimately AOT_have 1: x x ¬[E!]x]x
7645    by (metis "vdash-properties:6")
7646  AOT_hence x ¬[E!]x
7647    by (AOT_subst (reverse) ¬[E!]x  x ¬[E!]x]x bound: x)
7648       (auto intro!: "cqt:2" "beta-C-meta"[THEN "→E"])
7649  AOT_hence x ¬[E!]x by (metis "CBF" "vdash-properties:10")
7650  moreover AOT_obtain a where abs_a: O!a
7651    using "instantiation" "o-objects-exist:1" "qml:2" "vdash-properties:1[2]" "vdash-properties:6" by blast
7652  ultimately AOT_have ¬[E!]a using "∀E" by blast
7653  AOT_hence 2: ¬[E!]a by (metis "≡dfE" "conventions:5" "reductio-aa:1")
7654  AOT_have A!a
7655    apply (rule "=dfI"(2)[OF AOT_abstract])
7656     apply "cqt:2[lambda]"
7657    apply (rule "β←C"(1))
7658      apply "cqt:2[lambda]"
7659    using "cqt:2[const_var]"[axiom_inst] apply blast
7660    by (fact 2)
7661  AOT_thus p & ¬p for p using abs_a
7662    by (metis "≡E"(1) "oa-contingent:2" "reductio-aa:1")
7663qed
7664
7665AOT_theorem "prop-in-f:3:c": ¬Indiscriminate(O!)
7666proof(rule "raa-cor:2")
7667  AOT_assume Indiscriminate(O!)
7668  AOT_hence 0: (x O!x  x O!x)
7669    using "≡dfE"[OF "prop-indis"] "&E" by blast
7670  AOT_hence x O!x  x O!x
7671    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7672  moreover AOT_have x O!x
7673    using "o-objects-exist:1" by blast
7674  ultimately AOT_have x O!x
7675    by (metis "vdash-properties:6")
7676  AOT_thus p & ¬p for p
7677    by (metis "o-objects-exist:3" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7678qed
7679
7680AOT_theorem "prop-in-f:3:d": ¬Indiscriminate(A!)
7681proof(rule "raa-cor:2")
7682  AOT_assume Indiscriminate(A!)
7683  AOT_hence 0: (x A!x  x A!x)
7684    using "≡dfE"[OF "prop-indis"] "&E" by blast
7685  AOT_hence x A!x  x A!x
7686    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7687  moreover AOT_have x A!x
7688    using "o-objects-exist:2" by blast
7689  ultimately AOT_have x A!x
7690    by (metis "vdash-properties:6")
7691  AOT_thus p & ¬p for p
7692    by (metis "o-objects-exist:4" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7693qed
7694
7695AOT_theorem "prop-in-f:4:a": ¬Propositional(E!)
7696  using "modus-tollens:1" "prop-in-f:3:a" "prop-in-thm" by blast
7697
7698AOT_theorem "prop-in-f:4:b": ¬Propositional(E!-)
7699  using "modus-tollens:1" "prop-in-f:3:b" "prop-in-thm" by blast
7700
7701AOT_theorem "prop-in-f:4:c": ¬Propositional(O!)
7702  using "modus-tollens:1" "prop-in-f:3:c" "prop-in-thm" by blast
7703
7704AOT_theorem "prop-in-f:4:d": ¬Propositional(A!)
7705  using "modus-tollens:1" "prop-in-f:3:d" "prop-in-thm" by blast
7706
7707AOT_theorem "prop-prop-nec:1": p (F = y p])  p(F = y p])
7708proof(rule "→I")
7709  AOT_assume p (F = y p])
7710  AOT_hence p (F = y p])
7711    by (metis "BF◇" "vdash-properties:10")
7712  then AOT_obtain p where (F = y p]) using "∃E"[rotated] by blast
7713  AOT_hence F = y p] by (metis "derived-S5-rules:2" emptyE "id-nec:2" "vdash-properties:6")
7714  AOT_thus p(F = y p]) by (rule "∃I")
7715qed
7716
7717AOT_theorem "prop-prop-nec:2": p (F  y p])  p(F  y p])
7718proof(rule "→I")
7719  AOT_assume p (F  y p])
7720  AOT_hence (F  y p]) for p
7721    using "∀E" by blast
7722  AOT_hence (F  y p]) for p
7723    by (rule "id-nec2:2"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7724  AOT_hence p (F  y p]) by (rule GEN)
7725  AOT_thus p (F  y p]) using BF[THEN "→E"] by fast
7726qed
7727
7728AOT_theorem "prop-prop-nec:3": p (F = y p])  p(F = y p])
7729proof(rule "→I")
7730  AOT_assume p (F = y p])
7731  then AOT_obtain p where (F = y p]) using "∃E"[rotated] by blast
7732  AOT_hence (F = y p]) by (metis "id-nec:2" "vdash-properties:6")
7733  AOT_hence p(F = y p]) by (rule "∃I")
7734  AOT_thus p(F = y p]) by (metis Buridan "vdash-properties:10")
7735qed
7736
7737AOT_theorem "prop-prop-nec:4": p (F  y p])  p(F  y p])
7738proof(rule "→I")
7739  AOT_assume p (F  y p])
7740  AOT_hence p (F  y p]) by (metis "Buridan◇" "vdash-properties:10")
7741  AOT_hence (F  y p]) for p
7742    using "∀E" by blast
7743  AOT_hence F  y p] for p
7744    by (rule "id-nec2:3"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7745  AOT_thus p (F  y p]) by (rule GEN)
7746qed
7747
7748AOT_theorem "enc-prop-nec:1": F (x[F]  p(F = y p]))  F(x[F]  p (F = y p]))
7749proof(rule "→I"; rule GEN; rule "→I")
7750  fix F
7751  AOT_assume F (x[F]  p(F = y p]))
7752  AOT_hence F (x[F]  p(F = y p]))
7753    using "Buridan◇" "vdash-properties:10" by blast
7754  AOT_hence 0: (x[F]  p(F = y p])) using "∀E" by blast
7755  AOT_assume x[F]
7756  AOT_hence x[F] by (metis "en-eq:2[1]" "≡E"(1))
7757  AOT_hence p(F = y p])
7758    using 0 by (metis "KBasic2:4" "≡E"(1) "vdash-properties:10")
7759  AOT_thus p(F = y p])
7760    using "prop-prop-nec:1"[THEN "→E"] by blast
7761qed
7762
7763AOT_theorem "enc-prop-nec:2": F (x[F]  p(F = y p]))  F(x[F]  p (F = y p]))
7764  using "derived-S5-rules:1"[where Γ="{}", simplified, OF "enc-prop-nec:1"]
7765  by blast
7766
7767(*<*)
7768end
7769(*>*)